Calculus Examples

$lim_{x \to 0} \frac{\sin^{3} (x)}{\sin (x) - \tan (x)}$

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.

$\frac{lim_{x \to 0} \sin^{3} (x)}{lim_{x \to 0} \sin (x) - \tan (x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{{(lim_{x \to 0} \sin (x))}^{3}}{lim_{x \to 0} \sin (x) - \tan (x)}$

Step 1.1.2.1.2

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

$\frac{\sin^{3} (lim_{x \to 0} x)}{lim_{x \to 0} \sin (x) - \tan (x)}$

Step 1.1.2.2

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

$\frac{\sin^{3} (0)}{lim_{x \to 0} \sin (x) - \tan (x)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

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

$\frac{0^{3}}{lim_{x \to 0} \sin (x) - \tan (x)}$

Step 1.1.2.3.2

Raising $0$ to any positive power yields $0$ .

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

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 $x$ .

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

Step 1.1.3.4.2

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

$\frac{0}{\sin (0) - \tan (0)}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

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

$\frac{0}{0 - \tan (0)}$

Step 1.1.3.5.1.2

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

$\frac{0}{0 - 0}$

Step 1.1.3.5.1.3

Multiply $- 1$ by $0$ .

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

Step 1.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.5.3

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.

$lim_{x \to 0} \frac{\sin^{3} (x)}{\sin (x) - \tan (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin^{3} (x)]}{\frac{d}{d x} [\sin (x) - \tan (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [u^{3}] \frac{d}{d x} [\sin (x)]}{\frac{d}{d x} [\sin (x) - \tan (x)]}$

Step 1.3.2.2

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

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

Step 1.3.2.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

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

Step 2

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

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

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.

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

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

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

Step 3.1.2.5

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

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Step 3.1.2.5.1

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

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

Step 3.1.2.5.2

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

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

Step 3.1.2.6

Simplify the answer.

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Step 3.1.2.6.1

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

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

Step 3.1.2.6.2

Raising $0$ to any positive power yields $0$ .

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

Step 3.1.2.6.3

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

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

Step 3.1.2.6.4

Multiply $0$ by $1$ .

$3 \frac{0}{lim_{x \to 0} \cos (x) - \sec^{2} (x)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

$3 \frac{0}{lim_{x \to 0} \cos (x) - lim_{x \to 0} \sec^{2} (x)}$

Step 3.1.3.2

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

$3 \frac{0}{\cos (lim_{x \to 0} x) - lim_{x \to 0} \sec^{2} (x)}$

Step 3.1.3.3

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

$3 \frac{0}{\cos (lim_{x \to 0} x) - {(lim_{x \to 0} \sec (x))}^{2}}$

Step 3.1.3.4

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

$3 \frac{0}{\cos (lim_{x \to 0} x) - \sec^{2} (lim_{x \to 0} x)}$

Step 3.1.3.5

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

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Step 3.1.3.5.1

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

$3 \frac{0}{\cos (0) - \sec^{2} (lim_{x \to 0} x)}$

Step 3.1.3.5.2

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

$3 \frac{0}{\cos (0) - \sec^{2} (0)}$

Step 3.1.3.6

Simplify the answer.

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Step 3.1.3.6.1

Simplify each term.

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Step 3.1.3.6.1.1

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

$3 \frac{0}{1 - \sec^{2} (0)}$

Step 3.1.3.6.1.2

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

$3 \frac{0}{1 - 1^{2}}$

Step 3.1.3.6.1.3

One to any power is one.

$3 \frac{0}{1 - 1 \cdot 1}$

Step 3.1.3.6.1.4

Multiply $- 1$ by $1$ .

$3 \frac{0}{1 - 1}$

Step 3.1.3.6.2

Subtract $1$ from $1$ .

$3 (\frac{0}{0})$

Step 3.1.3.6.3

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

Undefined

$3 (\frac{0}{0})$

Step 3.1.3.7

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

Undefined

$3 (\frac{0}{0})$

Step 3.1.4

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

Undefined

$3 (\frac{0}{0})$

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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 $g (x) = \cos (x)$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

Move $\sin (x)$ .

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

Step 3.3.4.2

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

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Step 3.3.4.2.1

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

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

Step 3.3.4.2.2

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

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

Step 3.3.4.3

Add $1$ and $2$ .

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

Step 3.3.5

Move $- 1$ to the left of $\sin^{3} (x)$ .

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

Step 3.3.6

Rewrite $- 1 \sin^{3} (x)$ as $- \sin^{3} (x)$ .

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

Step 3.3.7

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 $g (x) = \sin (x)$ .

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Step 3.3.7.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 3.3.7.2

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

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

Step 3.3.7.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

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

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

Step 3.3.12

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

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

Step 3.3.13

Add $1$ and $1$ .

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

Step 3.3.14

Reorder terms.

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

Step 3.3.15

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

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

Step 3.3.16

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

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

Step 3.3.17

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

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Step 3.3.17.1

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

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

Step 3.3.17.2

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

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Step 3.3.17.2.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

Step 3.3.17.2.2

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

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

Step 3.3.17.2.3

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 3.3.17.3

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 3.3.17.4

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

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

Step 3.3.17.5

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

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

Step 3.3.17.6

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

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

Step 3.3.17.7

Add $1$ and $1$ .

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

Step 3.3.17.8

Multiply $2$ by $- 1$ .

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

Step 3.3.18

Reorder terms.

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

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

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

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

Step 4.1.2.5

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

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

Step 4.1.2.6

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

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

Step 4.1.2.7

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

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

Step 4.1.2.8

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

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

Step 4.1.2.9

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

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Step 4.1.2.9.1

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

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

Step 4.1.2.9.2

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

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

Step 4.1.2.9.3

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

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

Step 4.1.2.10

Simplify the answer.

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Step 4.1.2.10.1

Simplify each term.

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Step 4.1.2.10.1.1

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

$3 \frac{2 \cdot 1^{2} \cdot \sin (0) - \sin^{3} (0)}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.2

One to any power is one.

$3 \frac{2 \cdot 1 \cdot \sin (0) - \sin^{3} (0)}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.3

Multiply $2$ by $1$ .

$3 \frac{2 \cdot \sin (0) - \sin^{3} (0)}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.4

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

$3 \frac{2 \cdot 0 - \sin^{3} (0)}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.5

Multiply $2$ by $0$ .

$3 \frac{0 - \sin^{3} (0)}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.6

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

$3 \frac{0 - 0^{3}}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.7

Raising $0$ to any positive power yields $0$ .

$3 \frac{0 - 0}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.2.10.1.8

Multiply $- 1$ by $0$ .

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

Step 4.1.2.10.2

Add $0$ and $0$ .

$3 \frac{0}{lim_{x \to 0} - 2 \sec^{2} (x) \tan (x) - \sin (x)}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

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

$3 \frac{0}{- lim_{x \to 0} 2 \sec^{2} (x) \tan (x) - lim_{x \to 0} \sin (x)}$

Step 4.1.3.2

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

$3 \frac{0}{- 2 lim_{x \to 0} \sec^{2} (x) \tan (x) - lim_{x \to 0} \sin (x)}$

Step 4.1.3.3

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

$3 \frac{0}{- 2 lim_{x \to 0} \sec^{2} (x) \cdot lim_{x \to 0} \tan (x) - lim_{x \to 0} \sin (x)}$

Step 4.1.3.4

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

$3 \frac{0}{- 2 {(lim_{x \to 0} \sec (x))}^{2} \cdot lim_{x \to 0} \tan (x) - lim_{x \to 0} \sin (x)}$

Step 4.1.3.5

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

$3 \frac{0}{- 2 \sec^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \tan (x) - lim_{x \to 0} \sin (x)}$

Step 4.1.3.6

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

$3 \frac{0}{- 2 \sec^{2} (lim_{x \to 0} x) \cdot \tan (lim_{x \to 0} x) - lim_{x \to 0} \sin (x)}$

Step 4.1.3.7

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

$3 \frac{0}{- 2 \sec^{2} (lim_{x \to 0} x) \cdot \tan (lim_{x \to 0} x) - \sin (lim_{x \to 0} x)}$

Step 4.1.3.8

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

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Step 4.1.3.8.1

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

$3 \frac{0}{- 2 \sec^{2} (0) \cdot \tan (lim_{x \to 0} x) - \sin (lim_{x \to 0} x)}$

Step 4.1.3.8.2

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

$3 \frac{0}{- 2 \sec^{2} (0) \cdot \tan (0) - \sin (lim_{x \to 0} x)}$

Step 4.1.3.8.3

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

$3 \frac{0}{- 2 \sec^{2} (0) \cdot \tan (0) - \sin (0)}$

Step 4.1.3.9

Simplify the answer.

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Step 4.1.3.9.1

Simplify each term.

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Step 4.1.3.9.1.1

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

$3 \frac{0}{- 2 \cdot 1^{2} \cdot \tan (0) - \sin (0)}$

Step 4.1.3.9.1.2

One to any power is one.

$3 \frac{0}{- 2 \cdot 1 \cdot \tan (0) - \sin (0)}$

Step 4.1.3.9.1.3

Multiply $- 2$ by $1$ .

$3 \frac{0}{- 2 \cdot \tan (0) - \sin (0)}$

Step 4.1.3.9.1.4

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

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

Step 4.1.3.9.1.5

Multiply $- 2$ by $0$ .

$3 \frac{0}{0 - \sin (0)}$

Step 4.1.3.9.1.6

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

$3 \frac{0}{0 - 0}$

Step 4.1.3.9.1.7

Multiply $- 1$ by $0$ .

$3 \frac{0}{0 + 0}$

Step 4.1.3.9.2

Add $0$ and $0$ .

$3 (\frac{0}{0})$

Step 4.1.3.9.3

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

Undefined

$3 (\frac{0}{0})$

Step 4.1.3.10

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

Undefined

$3 (\frac{0}{0})$

Step 4.1.4

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

Undefined

$3 (\frac{0}{0})$

Step 4.2

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

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

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

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

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 $f (x) = \cos^{2} (x)$ and $g (x) = \sin (x)$ .

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

Step 4.3.3.3

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

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

Step 4.3.3.4

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 $g (x) = \cos (x)$ .

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Step 4.3.3.4.1

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

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

Step 4.3.3.4.2

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

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

Step 4.3.3.4.3

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

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

Step 4.3.3.5

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

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

Step 4.3.3.6

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

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Step 4.3.3.6.1

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

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Step 4.3.3.6.1.1

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

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

Step 4.3.3.6.1.2

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

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

Step 4.3.3.6.2

Add $2$ and $1$ .

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

Step 4.3.3.7

Multiply $- 1$ by $2$ .

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

Step 4.3.3.8

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

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

Step 4.3.3.9

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

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

Step 4.3.3.10

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

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

Step 4.3.3.11

Add $1$ and $1$ .

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

Step 4.3.4

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

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Step 4.3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin^{3} (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin^{3} (x)]$ .

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

Step 4.3.4.2

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

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Step 4.3.4.2.1

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

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

Step 4.3.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

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

Step 4.3.4.2.3

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

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

Step 4.3.4.3

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

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

Step 4.3.4.4

Multiply $3$ by $- 1$ .

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

Step 4.3.5

Simplify.

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Step 4.3.5.1

Apply the distributive property.

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

Step 4.3.5.2

Combine terms.

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Step 4.3.5.2.1

Multiply $- 2$ by $2$ .

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

Step 4.3.5.2.2

Reorder the factors of $- 4 \cos (x) \sin^{2} (x)$ .

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

Step 4.3.5.2.3

Subtract $3 \sin^{2} (x) \cos (x)$ from $- 4 \sin^{2} (x) \cos (x)$ .

$3 lim_{x \to 0} \frac{2 \cos^{3} (x) - 7 \sin^{2} (x) \cos (x)}{\frac{d}{d x} [- 2 \sec^{2} (x) \tan (x) - \sin (x)]}$

Step 4.3.5.3

Reorder terms.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{d}{d x} [- 2 \sec^{2} (x) \tan (x) - \sin (x)]}$

Step 4.3.6

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{d}{d x} [- 2 \sec^{2} (x) \tan (x)] + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7

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

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Step 4.3.7.1

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 \frac{d}{d x} [\sec^{2} (x) \tan (x)] + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.2

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 $f (x) = \sec^{2} (x)$ and $g (x) = \tan (x)$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{2} (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.3

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec^{2} (x)]) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.4

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 $g (x) = \sec (x)$ .

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Step 4.3.7.4.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) \frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.4.2

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) \cdot 2 u \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.4.3

Replace all occurrences of $u$ with $\sec (x)$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) \cdot 2 \sec (x) \frac{d}{d x} [\sec (x)]) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.5

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{2} (x) \sec^{2} (x) + \tan (x) \cdot 2 \sec (x) \sec (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.6

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

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Step 4.3.7.6.1

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 ({\sec (x)}^{2 + 2} + \tan (x) \cdot 2 \sec (x) \sec (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.6.2

Add $2$ and $2$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + \tan (x) \cdot 2 \sec (x) \sec (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.7

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + \tan (x) \cdot 2 \sec^{1} (x) \sec (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.8

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + \tan (x) \cdot 2 \sec^{1} (x) \sec^{1} (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.9

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + \tan (x) \cdot 2 {\sec (x)}^{1 + 1} \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.10

Add $1$ and $1$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + \tan (x) \cdot 2 \sec^{2} (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.11

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{1} (x) \tan (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.12

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{1} (x) \tan^{1} (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.13

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + 2 \sec^{2} (x) {\tan (x)}^{1 + 1}) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.7.14

Add $1$ and $1$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) + \frac{d}{d x} [- \sin (x)]}$

Step 4.3.8

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

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Step 4.3.8.1

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) - \frac{d}{d x} [\sin (x)]}$

Step 4.3.8.2

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 (\sec^{4} (x) + 2 \sec^{2} (x) \tan^{2} (x)) - \cos (x)}$

Step 4.3.9

Simplify.

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Step 4.3.9.1

Apply the distributive property.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 \sec^{4} (x) - 2 \cdot 2 \sec^{2} (x) \tan^{2} (x) - \cos (x)}$

Step 4.3.9.2

Multiply $2$ by $- 2$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 2 \sec^{4} (x) - 4 \sec^{2} (x) \tan^{2} (x) - \cos (x)}$

Step 4.3.9.3

Reorder terms.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 4 \sec^{2} (x) \tan^{2} (x) - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4

Simplify each term.

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Step 4.3.9.4.1

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 4 {(\frac{1}{\cos (x)})}^{2} \tan^{2} (x) - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.2

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 4 \frac{1^{2}}{\cos^{2} (x)} \tan^{2} (x) - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.3

One to any power is one.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- 4 \frac{1}{\cos^{2} (x)} \tan^{2} (x) - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.4

Combine $- 4$ and $\frac{1}{\cos^{2} (x)}$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{- 4}{\cos^{2} (x)} \tan^{2} (x) - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.5

Move the negative in front of the fraction.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4}{\cos^{2} (x)} \tan^{2} (x) - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.6

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4}{\cos^{2} (x)} {(\frac{\sin (x)}{\cos (x)})}^{2} - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.7

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4}{\cos^{2} (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.8

Multiply $- \frac{4}{\cos^{2} (x)} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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Step 4.3.9.4.8.1

Multiply $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ by $\frac{4}{\cos^{2} (x)}$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{\sin^{2} (x) \cdot 4}{\cos^{2} (x) \cos^{2} (x)} - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.8.2

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

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Step 4.3.9.4.8.2.1

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{\sin^{2} (x) \cdot 4}{{\cos (x)}^{2 + 2}} - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.8.2.2

Add $2$ and $2$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{\sin^{2} (x) \cdot 4}{\cos^{4} (x)} - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.9

Move $4$ to the left of $\sin^{2} (x)$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4 \cdot \sin^{2} (x)}{\cos^{4} (x)} - 2 \sec^{4} (x) - \cos (x)}$

Step 4.3.9.4.10

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4 \sin^{2} (x)}{\cos^{4} (x)} - 2 {(\frac{1}{\cos (x)})}^{4} - \cos (x)}$

Step 4.3.9.4.11

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

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4 \sin^{2} (x)}{\cos^{4} (x)} - 2 \frac{1^{4}}{\cos^{4} (x)} - \cos (x)}$

Step 4.3.9.4.12

One to any power is one.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4 \sin^{2} (x)}{\cos^{4} (x)} - 2 \frac{1}{\cos^{4} (x)} - \cos (x)}$

Step 4.3.9.4.13

Combine $- 2$ and $\frac{1}{\cos^{4} (x)}$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4 \sin^{2} (x)}{\cos^{4} (x)} + \frac{- 2}{\cos^{4} (x)} - \cos (x)}$

Step 4.3.9.4.14

Move the negative in front of the fraction.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{- \frac{4 \sin^{2} (x)}{\cos^{4} (x)} - \frac{2}{\cos^{4} (x)} - \cos (x)}$

Step 4.4

Combine terms.

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Step 4.4.1

Combine the numerators over the common denominator.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{- 4 \sin^{2} (x) - 2}{\cos^{4} (x)} - \cos (x)}$

Step 4.4.2

To write $- \cos (x)$ as a fraction with a common denominator, multiply by $\frac{\cos^{4} (x)}{\cos^{4} (x)}$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{- 4 \sin^{2} (x) - 2}{\cos^{4} (x)} - \cos (x) \cdot \frac{\cos^{4} (x)}{\cos^{4} (x)}}$

Step 4.4.3

Combine $- \cos (x)$ and $\frac{\cos^{4} (x)}{\cos^{4} (x)}$ .

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{- 4 \sin^{2} (x) - 2}{\cos^{4} (x)} + \frac{- \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 4.4.4

Combine the numerators over the common denominator.

$3 lim_{x \to 0} \frac{- 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{\frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5

Evaluate the limit.

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Step 5.1

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

$3 \frac{lim_{x \to 0} - 7 \sin^{2} (x) \cos (x) + 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.2

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

$3 \frac{- lim_{x \to 0} 7 \sin^{2} (x) \cos (x) + lim_{x \to 0} 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.3

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{- 7 lim_{x \to 0} \sin^{2} (x) \cos (x) + lim_{x \to 0} 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.4

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

$3 \frac{- 7 lim_{x \to 0} \sin^{2} (x) \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.5

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

$3 \frac{- 7 {(lim_{x \to 0} \sin (x))}^{2} \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.6

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (x) + lim_{x \to 0} 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.7

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + lim_{x \to 0} 2 \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.8

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 lim_{x \to 0} \cos^{3} (x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.9

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 {(lim_{x \to 0} \cos (x))}^{3}}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.10

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{lim_{x \to 0} \frac{- 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{\cos^{4} (x)}}$

Step 5.11

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{lim_{x \to 0} - 4 \sin^{2} (x) - 2 - \cos (x) \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.12

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- lim_{x \to 0} 4 \sin^{2} (x) - lim_{x \to 0} 2 - lim_{x \to 0} \cos (x) \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.13

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 lim_{x \to 0} \sin^{2} (x) - lim_{x \to 0} 2 - lim_{x \to 0} \cos (x) \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.14

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 {(lim_{x \to 0} \sin (x))}^{2} - lim_{x \to 0} 2 - lim_{x \to 0} \cos (x) \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.15

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - lim_{x \to 0} 2 - lim_{x \to 0} \cos (x) \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.16

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - lim_{x \to 0} \cos (x) \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.17

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - lim_{x \to 0} \cos (x) \cdot lim_{x \to 0} \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.18

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos^{4} (x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.19

Move the exponent $4$ from $\cos^{4} (x)$ outside the limit using the Limits Power Rule.

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot {(lim_{x \to 0} \cos (x))}^{4}}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.20

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

$3 \frac{- 7 \sin^{2} (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot \cos^{4} (lim_{x \to 0} x)}{lim_{x \to 0} \cos^{4} (x)}}$

Step 5.21

Move the exponent $4$ from $\cos^{4} (x)$ outside the limit using the Limits Power Rule.

Step 5.22

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

Step 6

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

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Step 6.1

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (lim_{x \to 0} x) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot \cos^{4} (lim_{x \to 0} x)}{\cos^{4} (lim_{x \to 0} x)}}$

Step 6.2

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (lim_{x \to 0} x)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot \cos^{4} (lim_{x \to 0} x)}{\cos^{4} (lim_{x \to 0} x)}}$

Step 6.3

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)}{\frac{- 4 \sin^{2} (lim_{x \to 0} x) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot \cos^{4} (lim_{x \to 0} x)}{\cos^{4} (lim_{x \to 0} x)}}$

Step 6.4

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)}{\frac{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (lim_{x \to 0} x) \cdot \cos^{4} (lim_{x \to 0} x)}{\cos^{4} (lim_{x \to 0} x)}}$

Step 6.5

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)}{\frac{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (lim_{x \to 0} x)}{\cos^{4} (lim_{x \to 0} x)}}$

Step 6.6

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)}{\frac{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)}{\cos^{4} (lim_{x \to 0} x)}}$

Step 6.7

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

$3 \frac{- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)}{\frac{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)}{\cos^{4} (0)}}$

Step 7

Simplify the answer.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{\cos^{4} (0)}{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.2

Simplify the numerator.

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Step 7.2.1

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

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1^{4}}{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.2.2

One to any power is one.

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 4 \sin^{2} (0) - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.3

Simplify the denominator.

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Step 7.3.1

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

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 4 \cdot 0^{2} - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.3.2

Raising $0$ to any positive power yields $0$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 4 \cdot 0 - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.3.3

Multiply $- 4$ by $0$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 1 \cdot 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.3.4

Multiply $- 1$ by $2$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - \cos (0) \cdot \cos^{4} (0)})$

Step 7.3.5

Multiply $\cos (0)$ by $\cos^{4} (0)$ by adding the exponents.

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Step 7.3.5.1

Move $\cos^{4} (0)$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - (\cos^{4} (0) \cos (0))})$

Step 7.3.5.2

Multiply $\cos^{4} (0)$ by $\cos (0)$ .

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Step 7.3.5.2.1

Raise $\cos (0)$ to the power of $1$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - (\cos^{4} (0) \cos^{1} (0))})$

Step 7.3.5.2.2

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

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - {\cos (0)}^{4 + 1}})$

Step 7.3.5.3

Add $4$ and $1$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - \cos^{5} (0)})$

Step 7.3.6

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

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - 1^{5}})$

Step 7.3.7

One to any power is one.

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - 1 \cdot 1})$

Step 7.3.8

Multiply $- 1$ by $1$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{0 - 2 - 1})$

Step 7.3.9

Subtract $2$ from $0$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 2 - 1})$

Step 7.3.10

Subtract $1$ from $- 2$ .

$3 ((- 7 \sin^{2} (0) \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 3})$

Step 7.4

Simplify each term.

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Step 7.4.1

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

$3 ((- 7 \cdot 0^{2} \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 3})$

Step 7.4.2

Raising $0$ to any positive power yields $0$ .

$3 ((- 7 \cdot 0 \cdot \cos (0) + 2 \cos^{3} (0)) \frac{1}{- 3})$

Step 7.4.3

Multiply $- 7$ by $0$ .

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

Step 7.4.4

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

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

Step 7.4.5

Multiply $0$ by $1$ .

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

Step 7.4.6

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

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

Step 7.4.7

One to any power is one.

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

Step 7.4.8

Multiply $2$ by $1$ .

$3 ((0 + 2) \frac{1}{- 3})$

Step 7.5

Add $0$ and $2$ .

$3 (2 (\frac{1}{- 3}))$

Step 7.6

Move the negative in front of the fraction.

$3 (2 (- \frac{1}{3}))$

Step 7.7

Multiply $2 (- \frac{1}{3})$ .

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Step 7.7.1

Multiply $- 1$ by $2$ .

$3 (- 2 (\frac{1}{3}))$

Step 7.7.2

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 7.8

Cancel the common factor of $3$ .

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Step 7.8.1

Cancel the common factor.

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

Step 7.8.2

Rewrite the expression.

$- 2$