Calculus Examples

$lim_{x \to 0} \frac{1 - \cos (2 x) + \tan^{2} (x)}{x \sin (2 x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} 1 - \cos (2 x) + \tan^{2} (x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 1 - lim_{x \to 0} \cos (2 x) + lim_{x \to 0} \tan^{2} (x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$\frac{1 - lim_{x \to 0} \cos (2 x) + lim_{x \to 0} \tan^{2} (x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.3

Move the limit inside the trig function because cosine is continuous.

$\frac{1 - \cos (lim_{x \to 0} 2 x) + lim_{x \to 0} \tan^{2} (x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1 - \cos (2 lim_{x \to 0} x) + lim_{x \to 0} \tan^{2} (x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.5

Move the exponent $2$ from $\tan^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{1 - \cos (2 lim_{x \to 0} x) + {(lim_{x \to 0} \tan (x))}^{2}}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.6

Move the limit inside the trig function because tangent is continuous.

$\frac{1 - \cos (2 lim_{x \to 0} x) + \tan^{2} (lim_{x \to 0} x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.7

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.1.2.7.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1 - \cos (2 \cdot 0) + \tan^{2} (lim_{x \to 0} x)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.7.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1 - \cos (2 \cdot 0) + \tan^{2} (0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Move $\tan^{2} (0)$ .

$\frac{1 + \tan^{2} (0) - \cos (2 \cdot 0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.2

Rearrange terms.

$\frac{\tan^{2} (0) + 1 - \cos (2 \cdot 0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.3

Apply pythagorean identity.

$\frac{\sec^{2} (0) - \cos (2 \cdot 0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.4

Simplify each term.

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Step 1.1.2.8.4.1

The exact value of $\sec (0)$ is $1$ .

$\frac{1^{2} - \cos (2 \cdot 0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.4.2

One to any power is one.

$\frac{1 - \cos (2 \cdot 0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.4.3

Multiply $2$ by $0$ .

$\frac{1 - \cos (0)}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.4.4

The exact value of $\cos (0)$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.4.5

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.2.8.5

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 0} x \sin (2 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{lim_{x \to 0} x \cdot lim_{x \to 0} \sin (2 x)}$

Step 1.1.3.2

Move the limit inside the trig function because sine is continuous.

$\frac{0}{lim_{x \to 0} x \cdot \sin (lim_{x \to 0} 2 x)}$

Step 1.1.3.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{lim_{x \to 0} x \cdot \sin (2 lim_{x \to 0} x)}$

Step 1.1.3.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.1.3.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0 \cdot \sin (2 lim_{x \to 0} x)}$

Step 1.1.3.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0 \cdot \sin (2 \cdot 0)}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Multiply $2$ by $0$ .

$\frac{0}{0 \cdot \sin (0)}$

Step 1.1.3.5.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{0 \cdot 0}$

Step 1.1.3.5.3

Multiply $0$ by $0$ .

$\frac{0}{0}$

Step 1.1.3.5.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{1 - \cos (2 x) + \tan^{2} (x)}{x \sin (2 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (2 x) + \tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [1 - \cos (2 x) + \tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.2

By the Sum Rule, the derivative of $1 - \cos (2 x) + \tan^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (2 x)] + \frac{d}{d x} [\tan^{2} (x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \cos (2 x)] + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{0 + \frac{d}{d x} [- \cos (2 x)] + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \cos (2 x)]$ .

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Step 1.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \cos (2 x)$ with respect to $x$ is $- \frac{d}{d x} [\cos (2 x)]$ .

$lim_{x \to 0} \frac{0 - \frac{d}{d x} [\cos (2 x)] + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 1.3.4.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$lim_{x \to 0} \frac{0 - (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$lim_{x \to 0} \frac{0 - (- \sin (u_{1}) \frac{d}{d x} [2 x]) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.2.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$lim_{x \to 0} \frac{0 - (- \sin (2 x) \frac{d}{d x} [2 x]) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.3

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{0 - (- \sin (2 x) (2 \frac{d}{d x} [x])) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{0 - (- \sin (2 x) (2 \cdot 1)) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.5

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{0 - (- \sin (2 x) \cdot 2) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.6

Multiply $2$ by $- 1$ .

$lim_{x \to 0} \frac{0 - (- 2 \sin (2 x)) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.4.7

Multiply $- 2$ by $- 1$ .

$lim_{x \to 0} \frac{0 + 2 \sin (2 x) + \frac{d}{d x} [\tan^{2} (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.5

Evaluate $\frac{d}{d x} [\tan^{2} (x)]$ .

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Step 1.3.5.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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Step 1.3.5.1.1

To apply the Chain Rule, set $u_{2}$ as $\tan (x)$ .

$lim_{x \to 0} \frac{0 + 2 \sin (2 x) + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.5.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{0 + 2 \sin (2 x) + 2 u_{2} \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.5.1.3

Replace all occurrences of $u_{2}$ with $\tan (x)$ .

$lim_{x \to 0} \frac{0 + 2 \sin (2 x) + 2 \tan (x) \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.5.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to 0} \frac{0 + 2 \sin (2 x) + 2 \tan (x) \sec^{2} (x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6

Simplify.

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Step 1.3.6.1

Add $0$ and $2 \sin (2 x)$ .

$lim_{x \to 0} \frac{2 \sin (2 x) + 2 \tan (x) \sec^{2} (x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.2

Reorder terms.

$lim_{x \to 0} \frac{2 \sec^{2} (x) \tan (x) + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3

Simplify each term.

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Step 1.3.6.3.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{2 {(\frac{1}{\cos (x)})}^{2} \tan (x) + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$lim_{x \to 0} \frac{2 \frac{1^{2}}{\cos^{2} (x)} \tan (x) + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.3

One to any power is one.

$lim_{x \to 0} \frac{2 \frac{1}{\cos^{2} (x)} \tan (x) + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.4

Combine $2$ and $\frac{1}{\cos^{2} (x)}$ .

$lim_{x \to 0} \frac{\frac{2}{\cos^{2} (x)} \tan (x) + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.5

Rewrite $\tan (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\frac{2}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.6

Combine.

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{2} (x) \cos (x)} + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.7

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 1.3.6.3.7.1

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 1.3.6.3.7.1.1

Raise $\cos (x)$ to the power of $1$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{2} (x) \cos^{1} (x)} + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.7.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{{\cos (x)}^{2 + 1}} + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.6.3.7.2

Add $2$ and $1$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{\frac{d}{d x} [x \sin (2 x)]}$

Step 1.3.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (2 x)$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x \frac{d}{d x} [\sin (2 x)] + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 1.3.8.1

To apply the Chain Rule, set $u$ as $2 x$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.8.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (\cos (u) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.8.3

Replace all occurrences of $u$ with $2 x$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (\cos (2 x) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.9

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (\cos (2 x) (2 \frac{d}{d x} [x])) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (\cos (2 x) (2 \cdot 1)) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.11

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (\cos (2 x) \cdot 2) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.12

Move $2$ to the left of $\cos (2 x)$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (2 \cdot \cos (2 x)) + \sin (2 x) \frac{d}{d x} [x]}$

Step 1.3.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (2 \cos (2 x)) + \sin (2 x) \cdot 1}$

Step 1.3.14

Multiply $\sin (2 x)$ by $1$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{x (2 \cos (2 x)) + \sin (2 x)}$

Step 1.3.15

Reorder terms.

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + 2 \sin (2 x)}{2 x \cos (2 x) + \sin (2 x)}$

Step 1.4

Combine terms.

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Step 1.4.1

To write $2 \sin (2 x)$ as a fraction with a common denominator, multiply by $\frac{\cos^{3} (x)}{\cos^{3} (x)}$ .

$lim_{x \to 0} \frac{\frac{2 \sin (x)}{\cos^{3} (x)} + \frac{2 \sin (2 x) \cos^{3} (x)}{\cos^{3} (x)}}{2 x \cos (2 x) + \sin (2 x)}$

Step 1.4.2

Combine the numerators over the common denominator.

$lim_{x \to 0} \frac{\frac{2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)}{\cos^{3} (x)}}{2 x \cos (2 x) + \sin (2 x)}$

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} \frac{2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)}{\cos^{3} (x)} \cdot \frac{1}{2 x \cos (2 x) + \sin (2 x)}$

Step 2.2

Multiply $\frac{2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)}{\cos^{3} (x)}$ by $\frac{1}{2 x \cos (2 x) + \sin (2 x)}$ .

$lim_{x \to 0} \frac{2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)}{\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 3.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} 2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} 2 \sin (x) + lim_{x \to 0} 2 \sin (2 x) \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.2

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to 0} \sin (x) + lim_{x \to 0} 2 \sin (2 x) \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.3

Move the limit inside the trig function because sine is continuous.

$\frac{2 \sin (lim_{x \to 0} x) + lim_{x \to 0} 2 \sin (2 x) \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \sin (lim_{x \to 0} x) + 2 lim_{x \to 0} \sin (2 x) \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.5

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{2 \sin (lim_{x \to 0} x) + 2 lim_{x \to 0} \sin (2 x) \cdot lim_{x \to 0} \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.6

Move the limit inside the trig function because sine is continuous.

$\frac{2 \sin (lim_{x \to 0} x) + 2 \sin (lim_{x \to 0} 2 x) \cdot lim_{x \to 0} \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \sin (lim_{x \to 0} x) + 2 \sin (2 lim_{x \to 0} x) \cdot lim_{x \to 0} \cos^{3} (x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.8

Move the exponent $3$ from $\cos^{3} (x)$ outside the limit using the Limits Power Rule.

$\frac{2 \sin (lim_{x \to 0} x) + 2 \sin (2 lim_{x \to 0} x) \cdot {(lim_{x \to 0} \cos (x))}^{3}}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.9

Move the limit inside the trig function because cosine is continuous.

$\frac{2 \sin (lim_{x \to 0} x) + 2 \sin (2 lim_{x \to 0} x) \cdot \cos^{3} (lim_{x \to 0} x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.10

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1.2.10.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \sin (0) + 2 \sin (2 lim_{x \to 0} x) \cdot \cos^{3} (lim_{x \to 0} x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.10.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \sin (0) + 2 \sin (2 \cdot 0) \cdot \cos^{3} (lim_{x \to 0} x)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.10.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{2 \sin (0) + 2 \sin (2 \cdot 0) \cdot \cos^{3} (0)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11

Simplify the answer.

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Step 3.1.2.11.1

Simplify each term.

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Step 3.1.2.11.1.1

The exact value of $\sin (0)$ is $0$ .

$\frac{2 \cdot 0 + 2 \sin (2 \cdot 0) \cdot \cos^{3} (0)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.2

Multiply $2$ by $0$ .

$\frac{0 + 2 \sin (2 \cdot 0) \cdot \cos^{3} (0)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.3

Multiply $2$ by $0$ .

$\frac{0 + 2 \sin (0) \cdot \cos^{3} (0)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.4

The exact value of $\sin (0)$ is $0$ .

$\frac{0 + 2 \cdot 0 \cdot \cos^{3} (0)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.5

Multiply $2$ by $0$ .

$\frac{0 + 0 \cdot \cos^{3} (0)}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.6

The exact value of $\cos (0)$ is $1$ .

$\frac{0 + 0 \cdot 1^{3}}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.7

One to any power is one.

$\frac{0 + 0 \cdot 1}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.1.8

Multiply $0$ by $1$ .

$\frac{0 + 0}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.2.11.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 0} \cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{lim_{x \to 0} \cos^{3} (x) \cdot lim_{x \to 0} 2 x \cos (2 x) + \sin (2 x)}$

Step 3.1.3.2

Move the exponent $3$ from $\cos^{3} (x)$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 0} \cos (x))}^{3} \cdot lim_{x \to 0} 2 x \cos (2 x) + \sin (2 x)}$

Step 3.1.3.3

Move the limit inside the trig function because cosine is continuous.

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot lim_{x \to 0} 2 x \cos (2 x) + \sin (2 x)}$

Step 3.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (lim_{x \to 0} 2 x \cos (2 x) + lim_{x \to 0} \sin (2 x))}$

Step 3.1.3.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (2 lim_{x \to 0} x \cos (2 x) + lim_{x \to 0} \sin (2 x))}$

Step 3.1.3.6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (2 lim_{x \to 0} x \cdot lim_{x \to 0} \cos (2 x) + lim_{x \to 0} \sin (2 x))}$

Step 3.1.3.7

Move the limit inside the trig function because cosine is continuous.

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (2 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} 2 x) + lim_{x \to 0} \sin (2 x))}$

Step 3.1.3.8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (2 lim_{x \to 0} x \cdot \cos (2 lim_{x \to 0} x) + lim_{x \to 0} \sin (2 x))}$

Step 3.1.3.9

Move the limit inside the trig function because sine is continuous.

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (2 lim_{x \to 0} x \cdot \cos (2 lim_{x \to 0} x) + \sin (lim_{x \to 0} 2 x))}$

Step 3.1.3.10

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{\cos^{3} (lim_{x \to 0} x) \cdot (2 lim_{x \to 0} x \cdot \cos (2 lim_{x \to 0} x) + \sin (2 lim_{x \to 0} x))}$

Step 3.1.3.11

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 3.1.3.11.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\cos^{3} (0) \cdot (2 lim_{x \to 0} x \cdot \cos (2 lim_{x \to 0} x) + \sin (2 lim_{x \to 0} x))}$

Step 3.1.3.11.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\cos^{3} (0) \cdot (2 \cdot 0 \cdot \cos (2 lim_{x \to 0} x) + \sin (2 lim_{x \to 0} x))}$

Step 3.1.3.11.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\cos^{3} (0) \cdot (2 \cdot 0 \cdot \cos (2 \cdot 0) + \sin (2 lim_{x \to 0} x))}$

Step 3.1.3.11.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\cos^{3} (0) \cdot (2 \cdot 0 \cdot \cos (2 \cdot 0) + \sin (2 \cdot 0))}$

Step 3.1.3.12

Simplify the answer.

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Step 3.1.3.12.1

The exact value of $\cos (0)$ is $1$ .

$\frac{0}{1^{3} \cdot (2 \cdot 0 \cdot \cos (2 \cdot 0) + \sin (2 \cdot 0))}$

Step 3.1.3.12.2

One to any power is one.

$\frac{0}{1 \cdot (2 \cdot 0 \cdot \cos (2 \cdot 0) + \sin (2 \cdot 0))}$

Step 3.1.3.12.3

Multiply $2 \cdot 0 \cdot \cos (2 \cdot 0) + \sin (2 \cdot 0)$ by $1$ .

$\frac{0}{2 \cdot 0 \cdot \cos (2 \cdot 0) + \sin (2 \cdot 0)}$

Step 3.1.3.12.4

Simplify each term.

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Step 3.1.3.12.4.1

Multiply $2$ by $0$ .

$\frac{0}{0 \cdot \cos (2 \cdot 0) + \sin (2 \cdot 0)}$

Step 3.1.3.12.4.2

Multiply $2$ by $0$ .

$\frac{0}{0 \cdot \cos (0) + \sin (2 \cdot 0)}$

Step 3.1.3.12.4.3

The exact value of $\cos (0)$ is $1$ .

$\frac{0}{0 \cdot 1 + \sin (2 \cdot 0)}$

Step 3.1.3.12.4.4

Multiply $0$ by $1$ .

$\frac{0}{0 + \sin (2 \cdot 0)}$

Step 3.1.3.12.4.5

Multiply $2$ by $0$ .

$\frac{0}{0 + \sin (0)}$

Step 3.1.3.12.4.6

The exact value of $\sin (0)$ is $0$ .

$\frac{0}{0 + 0}$

Step 3.1.3.12.5

Add $0$ and $0$ .

$\frac{0}{0}$

Step 3.1.3.12.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.3.13

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 3.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)}{\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))} = lim_{x \to 0} \frac{\frac{d}{d x} [2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)]}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)]}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.2

By the Sum Rule, the derivative of $2 \sin (x) + 2 \sin (2 x) \cos^{3} (x)$ with respect to $x$ is $\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [2 \sin (2 x) \cos^{3} (x)]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [2 \sin (x)] + \frac{d}{d x} [2 \sin (2 x) \cos^{3} (x)]}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.3

Evaluate $\frac{d}{d x} [2 \sin (x)]$ .

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Step 3.3.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (x)]$ .

$lim_{x \to 0} \frac{2 \frac{d}{d x} [\sin (x)] + \frac{d}{d x} [2 \sin (2 x) \cos^{3} (x)]}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.3.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + \frac{d}{d x} [2 \sin (2 x) \cos^{3} (x)]}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4

Evaluate $\frac{d}{d x} [2 \sin (2 x) \cos^{3} (x)]$ .

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Step 3.3.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin (2 x) \cos^{3} (x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin (2 x) \cos^{3} (x)]$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 \frac{d}{d x} [\sin (2 x) \cos^{3} (x)]}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (2 x)$ and $g (x) = \cos^{3} (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) \frac{d}{d x} [\cos^{3} (x)] + \cos^{3} (x) \frac{d}{d x} [\sin (2 x)])}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

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Step 3.3.4.3.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\cos (x)]) + \cos^{3} (x) \frac{d}{d x} [\sin (2 x)])}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 {u_{1}}^{2} \frac{d}{d x} [\cos (x)]) + \cos^{3} (x) \frac{d}{d x} [\sin (2 x)])}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.3.3

Replace all occurrences of $u_{1}$ with $\cos (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) \frac{d}{d x} [\cos (x)]) + \cos^{3} (x) \frac{d}{d x} [\sin (2 x)])}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.4

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) (- \sin (x))) + \cos^{3} (x) \frac{d}{d x} [\sin (2 x)])}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 3.3.4.5.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) (- \sin (x))) + \cos^{3} (x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [2 x]))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.5.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) (- \sin (x))) + \cos^{3} (x) (\cos (u_{2}) \frac{d}{d x} [2 x]))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.5.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) (- \sin (x))) + \cos^{3} (x) (\cos (2 x) \frac{d}{d x} [2 x]))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.6

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) (- \sin (x))) + \cos^{3} (x) (\cos (2 x) (2 \frac{d}{d x} [x])))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (3 \cos^{2} (x) (- \sin (x))) + \cos^{3} (x) (\cos (2 x) (2 \cdot 1)))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.8

Multiply $- 1$ by $3$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (- 3 \cos^{2} (x) \sin (x)) + \cos^{3} (x) (\cos (2 x) (2 \cdot 1)))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.9

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (- 3 \cos^{2} (x) \sin (x)) + \cos^{3} (x) (\cos (2 x) \cdot 2))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.10

Move $2$ to the left of $\cos (2 x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (- 3 \cos^{2} (x) \sin (x)) + \cos^{3} (x) (2 \cdot \cos (2 x)))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.4.11

Move $2$ to the left of $\cos^{3} (x)$ .

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (- 3 \cos^{2} (x) \sin (x)) + 2 \cos^{3} (x) \cos (2 x))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.5

Simplify.

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Step 3.3.5.1

Apply the distributive property.

$lim_{x \to 0} \frac{2 \cos (x) + 2 (\sin (2 x) (- 3 \cos^{2} (x) \sin (x))) + 2 (2 \cos^{3} (x) \cos (2 x))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.5.2

Combine terms.

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Step 3.3.5.2.1

Multiply $- 3$ by $2$ .

$lim_{x \to 0} \frac{2 \cos (x) - 6 (\sin (2 x) (\cos^{2} (x) \sin (x))) + 2 (2 \cos^{3} (x) \cos (2 x))}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.5.2.2

Multiply $2$ by $2$ .

$lim_{x \to 0} \frac{2 \cos (x) - 6 \sin (2 x) \cos^{2} (x) \sin (x) + 4 \cos^{3} (x) \cos (2 x)}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.5.3

Reorder terms.

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\frac{d}{d x} [\cos^{3} (x) (2 x \cos (2 x) + \sin (2 x))]}$

Step 3.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \cos^{3} (x)$ and $g (x) = 2 x \cos (2 x) + \sin (2 x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) \frac{d}{d x} [2 x \cos (2 x) + \sin (2 x)] + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.7

By the Sum Rule, the derivative of $2 x \cos (2 x) + \sin (2 x)$ with respect to $x$ is $\frac{d}{d x} [2 x \cos (2 x)] + \frac{d}{d x} [\sin (2 x)]$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (\frac{d}{d x} [2 x \cos (2 x)] + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.8

Since $2$ is constant with respect to $x$ , the derivative of $2 x \cos (2 x)$ with respect to $x$ is $2 \frac{d}{d x} [x \cos (2 x)]$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 \frac{d}{d x} [x \cos (2 x)] + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.9

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (2 x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x \frac{d}{d x} [\cos (2 x)] + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.10

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 3.3.10.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.10.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- \sin (u_{1}) \frac{d}{d x} [2 x]) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.10.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- \sin (2 x) \frac{d}{d x} [2 x]) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.11

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- \sin (2 x) (2 \frac{d}{d x} [x])) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.12

Multiply $2$ by $- 1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x) \frac{d}{d x} [x]) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x) \cdot 1) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.14

Multiply $- 2$ by $1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x) \frac{d}{d x} [x]) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x) \cdot 1) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.16

Multiply $\cos (2 x)$ by $1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \frac{d}{d x} [\sin (2 x)]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.17

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 3.3.17.1

To apply the Chain Rule, set $u_{2}$ as $2 x$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [2 x]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.17.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \cos (u_{2}) \frac{d}{d x} [2 x]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.17.3

Replace all occurrences of $u_{2}$ with $2 x$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \cos (2 x) \frac{d}{d x} [2 x]) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.18

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \cos (2 x) (2 \frac{d}{d x} [x])) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.19

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \cos (2 x) (2 \cdot 1)) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.20

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + \cos (2 x) \cdot 2) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.21

Move $2$ to the left of $\cos (2 x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + 2 \cdot \cos (2 x)) + (2 x \cos (2 x) + \sin (2 x)) \frac{d}{d x} [\cos^{3} (x)]}$

Step 3.3.22

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (x)$ .

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Step 3.3.22.1

To apply the Chain Rule, set $u_{3}$ as $\cos (x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + 2 \cos (2 x)) + (2 x \cos (2 x) + \sin (2 x)) (\frac{d}{d u_{3}} [{u_{3}}^{3}] \frac{d}{d x} [\cos (x)])}$

Step 3.3.22.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 3$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + 2 \cos (2 x)) + (2 x \cos (2 x) + \sin (2 x)) (3 {u_{3}}^{2} \frac{d}{d x} [\cos (x)])}$

Step 3.3.22.3

Replace all occurrences of $u_{3}$ with $\cos (x)$ .

Step 3.3.23

Move $3$ to the left of $2 x \cos (2 x) + \sin (2 x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + 2 \cos (2 x)) + 3 \cdot (2 x \cos (2 x) + \sin (2 x)) \cos^{2} (x) \frac{d}{d x} [\cos (x)]}$

Step 3.3.24

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + 2 \cos (2 x)) + 3 (2 x \cos (2 x) + \sin (2 x)) \cos^{2} (x) (- \sin (x))}$

Step 3.3.25

Multiply $- 1$ by $3$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)) + \cos (2 x)) + 2 \cos (2 x)) - 3 (2 x \cos (2 x) + \sin (2 x)) \cos^{2} (x) \sin (x)}$

Step 3.3.26

Simplify.

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Step 3.3.26.1

Apply the distributive property.

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x))) + 2 \cos (2 x) + 2 \cos (2 x)) - 3 (2 x \cos (2 x) + \sin (2 x)) \cos^{2} (x) \sin (x)}$

Step 3.3.26.2

Apply the distributive property.

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)))) + \cos^{3} (x) (2 \cos (2 x)) + \cos^{3} (x) (2 \cos (2 x)) - 3 (2 x \cos (2 x) + \sin (2 x)) \cos^{2} (x) \sin (x)}$

Step 3.3.26.3

Apply the distributive property.

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)))) + \cos^{3} (x) (2 \cos (2 x)) + \cos^{3} (x) (2 \cos (2 x)) + (- 3 (2 x \cos (2 x)) - 3 \sin (2 x)) \cos^{2} (x) \sin (x)}$

Step 3.3.26.4

Apply the distributive property.

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (2 (x (- 2 \sin (2 x)))) + \cos^{3} (x) (2 \cos (2 x)) + \cos^{3} (x) (2 \cos (2 x)) + (- 3 (2 x \cos (2 x)) \cos^{2} (x) - 3 \sin (2 x) \cos^{2} (x)) \sin (x)}$

Step 3.3.26.5

Apply the distributive property.

Step 3.3.26.6

Combine terms.

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Step 3.3.26.6.1

Multiply $- 2$ by $2$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (- 4 (x (\sin (2 x)))) + \cos^{3} (x) (2 \cos (2 x)) + \cos^{3} (x) (2 \cos (2 x)) - 3 (2 x \cos (2 x)) \cos^{2} (x) \sin (x) - 3 \sin (2 x) \cos^{2} (x) \sin (x)}$

Step 3.3.26.6.2

Move $2$ to the left of $\cos^{3} (x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (- 4 x \sin (2 x)) + 2 \cdot \cos^{3} (x) \cos (2 x) + \cos^{3} (x) (2 \cos (2 x)) - 3 (2 x \cos (2 x)) \cos^{2} (x) \sin (x) - 3 \sin (2 x) \cos^{2} (x) \sin (x)}$

Step 3.3.26.6.3

Move $2$ to the left of $\cos^{3} (x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (- 4 x \sin (2 x)) + 2 \cos^{3} (x) \cos (2 x) + 2 \cdot \cos^{3} (x) \cos (2 x) - 3 (2 x \cos (2 x)) \cos^{2} (x) \sin (x) - 3 \sin (2 x) \cos^{2} (x) \sin (x)}$

Step 3.3.26.6.4

Add $2 \cos^{3} (x) \cos (2 x)$ and $2 \cos^{3} (x) \cos (2 x)$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (- 4 x \sin (2 x)) + 4 \cos^{3} (x) \cos (2 x) - 3 (2 x \cos (2 x)) \cos^{2} (x) \sin (x) - 3 \sin (2 x) \cos^{2} (x) \sin (x)}$

Step 3.3.26.6.5

Multiply $2$ by $- 3$ .

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{\cos^{3} (x) (- 4 x \sin (2 x)) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos (2 x) \cos^{2} (x) \sin (x) - 3 \sin (2 x) \cos^{2} (x) \sin (x)}$

Step 3.3.26.7

Reorder terms.

$lim_{x \to 0} \frac{- 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{- 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4

Evaluate the limit.

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Step 4.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} - 6 \cos^{2} (x) \sin (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) + 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- lim_{x \to 0} 6 \cos^{2} (x) \sin (x) \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 lim_{x \to 0} \cos^{2} (x) \sin (x) \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.4

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 lim_{x \to 0} \cos^{2} (x) \cdot lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.5

Move the exponent $2$ from $\cos^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{- 6 {(lim_{x \to 0} \cos (x))}^{2} \cdot lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.6

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.7

Move the limit inside the trig function because sine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.8

Move the limit inside the trig function because sine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.9

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.10

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos^{3} (x) \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.11

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos^{3} (x) \cdot lim_{x \to 0} \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.12

Move the exponent $3$ from $\cos^{3} (x)$ outside the limit using the Limits Power Rule.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 {(lim_{x \to 0} \cos (x))}^{3} \cdot lim_{x \to 0} \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.13

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.14

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 2 x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.15

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + lim_{x \to 0} 2 \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.16

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 lim_{x \to 0} \cos (x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.17

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{lim_{x \to 0} - 4 x \cos^{3} (x) \sin (2 x) + 4 \cos^{3} (x) \cos (2 x) - 6 x \cos^{2} (x) \cos (2 x) \sin (x) - 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.18

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- lim_{x \to 0} 4 x \cos^{3} (x) \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.19

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cos^{3} (x) \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.20

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot lim_{x \to 0} \cos^{3} (x) \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.21

Move the exponent $3$ from $\cos^{3} (x)$ outside the limit using the Limits Power Rule.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot {(lim_{x \to 0} \cos (x))}^{3} \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.22

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.23

Move the limit inside the trig function because sine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 2 x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.24

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + lim_{x \to 0} 4 \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.25

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos^{3} (x) \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.26

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 lim_{x \to 0} \cos^{3} (x) \cdot lim_{x \to 0} \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.27

Move the exponent $3$ from $\cos^{3} (x)$ outside the limit using the Limits Power Rule.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 {(lim_{x \to 0} \cos (x))}^{3} \cdot lim_{x \to 0} \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.28

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.29

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 2 x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.30

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - lim_{x \to 0} 6 x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.31

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cos^{2} (x) \cos (2 x) \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.32

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot lim_{x \to 0} \cos^{2} (x) \cdot lim_{x \to 0} \cos (2 x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.33

Move the exponent $2$ from $\cos^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot {(lim_{x \to 0} \cos (x))}^{2} \cdot lim_{x \to 0} \cos (2 x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.34

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (2 x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.35

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} 2 x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.36

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.37

Move the limit inside the trig function because sine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - lim_{x \to 0} 3 \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.38

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 lim_{x \to 0} \cos^{2} (x) \sin (x) \sin (2 x)}$

Step 4.39

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 lim_{x \to 0} \cos^{2} (x) \cdot lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (2 x)}$

Step 4.40

Move the exponent $2$ from $\cos^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 {(lim_{x \to 0} \cos (x))}^{2} \cdot lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (2 x)}$

Step 4.41

Move the limit inside the trig function because cosine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \sin (2 x)}$

Step 4.42

Move the limit inside the trig function because sine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (2 x)}$

Step 4.43

Move the limit inside the trig function because sine is continuous.

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 2 x)}$

Step 4.44

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 6 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

Tap for more steps...

Step 5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 lim_{x \to 0} x) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (lim_{x \to 0} x)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.6

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 lim_{x \to 0} x \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.7

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.8

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 lim_{x \to 0} x) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.9

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.10

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 lim_{x \to 0} x) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.11

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 lim_{x \to 0} x \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.12

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (lim_{x \to 0} x) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.13

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.14

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (lim_{x \to 0} x) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.15

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.16

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (lim_{x \to 0} x) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.17

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 lim_{x \to 0} x)}$

Step 5.18

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{- 6 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

The exact value of $\cos (0)$ is $1$ .

$\frac{- 6 \cdot 1^{2} \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.2

One to any power is one.

$\frac{- 6 \cdot 1 \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.3

Multiply $- 6$ by $1$ .

$\frac{- 6 \cdot \sin (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.4

The exact value of $\sin (0)$ is $0$ .

$\frac{- 6 \cdot 0 \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.5

Multiply $- 6$ by $0$ .

$\frac{0 \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.6

Multiply $2$ by $0$ .

$\frac{0 \cdot \sin (0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.7

The exact value of $\sin (0)$ is $0$ .

$\frac{0 \cdot 0 + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.8

Multiply $0$ by $0$ .

$\frac{0 + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.9

The exact value of $\cos (0)$ is $1$ .

$\frac{0 + 4 \cdot 1^{3} \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.10

One to any power is one.

$\frac{0 + 4 \cdot 1 \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.11

Multiply $4$ by $1$ .

$\frac{0 + 4 \cdot \cos (2 \cdot 0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.12

Multiply $2$ by $0$ .

$\frac{0 + 4 \cdot \cos (0) + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.13

The exact value of $\cos (0)$ is $1$ .

$\frac{0 + 4 \cdot 1 + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.14

Multiply $4$ by $1$ .

$\frac{0 + 4 + 2 \cos (0)}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.15

The exact value of $\cos (0)$ is $1$ .

$\frac{0 + 4 + 2 \cdot 1}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.16

Multiply $2$ by $1$ .

$\frac{0 + 4 + 2}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.17

Add $0$ and $4$ .

$\frac{4 + 2}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.1.18

Add $4$ and $2$ .

$\frac{6}{- 4 \cdot 0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 4$ by $0$ .

$\frac{6}{0 \cdot \cos^{3} (0) \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.2

The exact value of $\cos (0)$ is $1$ .

$\frac{6}{0 \cdot 1^{3} \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.3

One to any power is one.

$\frac{6}{0 \cdot 1 \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.4

Multiply $0$ by $1$ .

$\frac{6}{0 \cdot \sin (2 \cdot 0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.5

Multiply $2$ by $0$ .

$\frac{6}{0 \cdot \sin (0) + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.6

The exact value of $\sin (0)$ is $0$ .

$\frac{6}{0 \cdot 0 + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.7

Multiply $0$ by $0$ .

$\frac{6}{0 + 4 \cos^{3} (0) \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.8

The exact value of $\cos (0)$ is $1$ .

$\frac{6}{0 + 4 \cdot 1^{3} \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.9

One to any power is one.

$\frac{6}{0 + 4 \cdot 1 \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.10

Multiply $4$ by $1$ .

$\frac{6}{0 + 4 \cdot \cos (2 \cdot 0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.11

Multiply $2$ by $0$ .

$\frac{6}{0 + 4 \cdot \cos (0) - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.12

The exact value of $\cos (0)$ is $1$ .

$\frac{6}{0 + 4 \cdot 1 - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.13

Multiply $4$ by $1$ .

$\frac{6}{0 + 4 - 6 \cdot 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.14

Multiply $- 6$ by $0$ .

$\frac{6}{0 + 4 + 0 \cdot \cos^{2} (0) \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.15

The exact value of $\cos (0)$ is $1$ .

$\frac{6}{0 + 4 + 0 \cdot 1^{2} \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.16

One to any power is one.

$\frac{6}{0 + 4 + 0 \cdot 1 \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.17

Multiply $0$ by $1$ .

$\frac{6}{0 + 4 + 0 \cdot \cos (2 \cdot 0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.18

Multiply $2$ by $0$ .

$\frac{6}{0 + 4 + 0 \cdot \cos (0) \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.19

The exact value of $\cos (0)$ is $1$ .

$\frac{6}{0 + 4 + 0 \cdot 1 \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.20

Multiply $0$ by $1$ .

$\frac{6}{0 + 4 + 0 \cdot \sin (0) - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.21

The exact value of $\sin (0)$ is $0$ .

$\frac{6}{0 + 4 + 0 \cdot 0 - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.22

Multiply $0$ by $0$ .

$\frac{6}{0 + 4 + 0 - 3 \cos^{2} (0) \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.23

The exact value of $\cos (0)$ is $1$ .

$\frac{6}{0 + 4 + 0 - 3 \cdot 1^{2} \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.24

One to any power is one.

$\frac{6}{0 + 4 + 0 - 3 \cdot 1 \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.25

Multiply $- 3$ by $1$ .

$\frac{6}{0 + 4 + 0 - 3 \cdot \sin (0) \cdot \sin (2 \cdot 0)}$

Step 6.2.26

The exact value of $\sin (0)$ is $0$ .

$\frac{6}{0 + 4 + 0 - 3 \cdot 0 \cdot \sin (2 \cdot 0)}$

Step 6.2.27

Multiply $- 3$ by $0$ .

$\frac{6}{0 + 4 + 0 + 0 \cdot \sin (2 \cdot 0)}$

Step 6.2.28

Multiply $2$ by $0$ .

$\frac{6}{0 + 4 + 0 + 0 \cdot \sin (0)}$

Step 6.2.29

The exact value of $\sin (0)$ is $0$ .

$\frac{6}{0 + 4 + 0 + 0 \cdot 0}$

Step 6.2.30

Multiply $0$ by $0$ .

$\frac{6}{0 + 4 + 0 + 0}$

Step 6.2.31

Add $0 + 4 + 0$ and $0$ .

$\frac{6}{0 + 4 + 0}$

Step 6.2.32

Add $0 + 4$ and $0$ .

$\frac{6}{0 + 4}$

Step 6.2.33

Add $0$ and $4$ .

$\frac{6}{4}$

Step 6.3

Cancel the common factor of $6$ and $4$ .

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Step 6.3.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4}$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 3}{2 \cdot 2}$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 2}$

Step 6.3.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$