Calculus Examples

$lim_{x \to 0} \frac{\ln (\cos (4 x))}{3 x^{2}}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\ln (\cos (4 x))}{x^{2}}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{1}{3} \cdot \frac{lim_{x \to 0} \ln (\cos (4 x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Move the limit inside the logarithm.

$\frac{1}{3} \cdot \frac{\ln (lim_{x \to 0} \cos (4 x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.1.2

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

$\frac{1}{3} \cdot \frac{\ln (\cos (lim_{x \to 0} 4 x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.1.3

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

$\frac{1}{3} \cdot \frac{\ln (\cos (4 lim_{x \to 0} x))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.2

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

$\frac{1}{3} \cdot \frac{\ln (\cos (4 \cdot 0))}{lim_{x \to 0} x^{2}}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $4$ by $0$ .

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

Step 2.1.2.3.2

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

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

Step 2.1.2.3.3

The natural logarithm of $1$ is $0$ .

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x^{2}}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

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

Step 2.1.3.4

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

Undefined

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

Step 2.1.4

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

Undefined

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

Step 2.2

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.

$lim_{x \to 0} \frac{\ln (\cos (4 x))}{x^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\cos (4 x))]}{\frac{d}{d x} [x^{2}]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\cos (4 x))]}{\frac{d}{d x} [x^{2}]}$

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \cos (4 x)$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (4 x)$ .

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

Step 2.3.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 2.3.2.3

Replace all occurrences of $u_{1}$ with $\cos (4 x)$ .

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

Step 2.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $4 x$ .

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $4 x$ .

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

Step 2.3.4

Combine $\frac{1}{\cos (4 x)}$ and $\sin (4 x)$ .

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

Step 2.3.5

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.3.6

Multiply $4$ by $- 1$ .

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

Step 2.3.7

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

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

Step 2.3.8

Move the negative in front of the fraction.

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

Step 2.3.9

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

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

Step 2.3.10

Multiply $- 1$ by $1$ .

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

Step 2.3.11

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

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

Multiply $\frac{1}{2 x}$ by $\frac{4 \sin (4 x)}{\cos (4 x)}$ .

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

Step 2.6

Cancel the common factor of $4$ and $2$ .

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Step 2.6.1

Factor $2$ out of $4 \sin (4 x)$ .

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

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Cancel the common factor.

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

Step 2.6.2.2

Rewrite the expression.

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

Step 3

Evaluate the limit.

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Step 3.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 3.2

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

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

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.

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

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

Evaluate the limit.

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Step 4.1.2.1.1

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

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

Step 4.1.2.1.2

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Simplify the answer.

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Step 4.1.2.3.1

Multiply $4$ by $0$ .

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

Step 4.1.2.3.2

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

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

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Step 4.1.3.4.1

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

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

Step 4.1.3.4.2

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

$\frac{1}{3} \cdot - 1 \cdot 2 \frac{0}{0 \cdot \cos (4 \cdot 0)}$

Step 4.1.3.5

Simplify the answer.

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Step 4.1.3.5.1

Multiply $4$ by $0$ .

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

Step 4.1.3.5.2

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

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

Step 4.1.3.5.3

Multiply $0$ by $1$ .

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

Step 4.1.3.5.4

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

Undefined

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

Step 4.1.3.6

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

Undefined

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

Step 4.1.4

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

Undefined

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

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.

$lim_{x \to 0} \frac{\sin (4 x)}{x \cos (4 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [x \cos (4 x)]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 4 x$ .

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Step 4.3.2.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 4.3.2.2

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

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

Step 4.3.2.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 4.3.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 4.3.4

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

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

Step 4.3.5

Multiply $4$ by $1$ .

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

Step 4.3.6

Move $4$ to the left of $\cos (4 x)$ .

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

Step 4.3.7

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 $g (x) = \cos (4 x)$ .

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

Step 4.3.8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 x$ .

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Step 4.3.8.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 4.3.8.2

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

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

Step 4.3.8.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 4.3.9

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 4.3.10

Multiply $4$ by $- 1$ .

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

Step 4.3.11

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

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

Step 4.3.12

Multiply $- 4$ by $1$ .

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

Step 4.3.13

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

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

Step 4.3.14

Multiply $\cos (4 x)$ by $1$ .

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

Step 4.3.15

Reorder terms.

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

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 $x$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Move the negative in front of the fraction.

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

Step 7.3

Multiply $- \frac{1}{3} \cdot 2$ .

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Step 7.3.1

Multiply $2$ by $- 1$ .

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

Step 7.3.2

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

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

Step 7.4

Move the negative in front of the fraction.

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

Step 7.5

Multiply $- \frac{2}{3} \cdot 4$ .

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Step 7.5.1

Multiply $4$ by $- 1$ .

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

Step 7.5.2

Combine $- 4$ and $\frac{2}{3}$ .

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

Step 7.5.3

Multiply $- 4$ by $2$ .

$\frac{- 8}{3} \cdot \frac{\cos (4 \cdot 0)}{- 4 \cdot 0 \cdot \sin (4 \cdot 0) + \cos (4 \cdot 0)}$

Step 7.6

Move the negative in front of the fraction.

$- \frac{8}{3} \cdot \frac{\cos (4 \cdot 0)}{- 4 \cdot 0 \cdot \sin (4 \cdot 0) + \cos (4 \cdot 0)}$

Step 7.7

Simplify the numerator.

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Step 7.7.1

Multiply $4$ by $0$ .

$- \frac{8}{3} \cdot \frac{\cos (0)}{- 4 \cdot 0 \cdot \sin (4 \cdot 0) + \cos (4 \cdot 0)}$

Step 7.7.2

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

$- \frac{8}{3} \cdot \frac{1}{- 4 \cdot 0 \cdot \sin (4 \cdot 0) + \cos (4 \cdot 0)}$

Step 7.8

Simplify the denominator.

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Step 7.8.1

Multiply $- 4$ by $0$ .

$- \frac{8}{3} \cdot \frac{1}{0 \cdot \sin (4 \cdot 0) + \cos (4 \cdot 0)}$

Step 7.8.2

Multiply $4$ by $0$ .

$- \frac{8}{3} \cdot \frac{1}{0 \cdot \sin (0) + \cos (4 \cdot 0)}$

Step 7.8.3

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

$- \frac{8}{3} \cdot \frac{1}{0 \cdot 0 + \cos (4 \cdot 0)}$

Step 7.8.4

Multiply $0$ by $0$ .

$- \frac{8}{3} \cdot \frac{1}{0 + \cos (4 \cdot 0)}$

Step 7.8.5

Multiply $4$ by $0$ .

$- \frac{8}{3} \cdot \frac{1}{0 + \cos (0)}$

Step 7.8.6

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

$- \frac{8}{3} \cdot \frac{1}{0 + 1}$

Step 7.8.7

Add $0$ and $1$ .

$- \frac{8}{3} \cdot \frac{1}{1}$

Step 7.9

Cancel the common factor of $1$ .

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Step 7.9.1

Cancel the common factor.

$- \frac{8}{3} \cdot \frac{1}{1}$

Step 7.9.2

Rewrite the expression.

$- \frac{8}{3} \cdot 1$

Step 7.10

Multiply $- 1$ by $1$ .

$- \frac{8}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{3}$

Decimal Form:

$- 2.\overline{6}$