Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Move the limit inside the logarithm.

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the limit inside the logarithm.

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.2

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

$\frac{0}{\ln (\cos (3 \cdot 0))}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $3$ by $0$ .

$\frac{0}{\ln (\cos (0))}$

Step 1.3.3.2

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

$\frac{0}{\ln (1)}$

Step 1.3.3.3

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

$\frac{0}{0}$

Step 1.3.3.4

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{\ln (\cos (x))}{\ln (\cos (3 x))} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (\cos (x))]}{\frac{d}{d x} [\ln (\cos (3 x))]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

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

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Step 3.2.1

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

$lim_{x \to 0} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\cos (x)]}{\frac{d}{d x} [\ln (\cos (3 x))]}$

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

To apply the Chain Rule, set $u_{1}$ as $\cos (3 x)$ .

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

Step 3.5.2

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

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

Step 3.5.3

Replace all occurrences of $u_{1}$ with $\cos (3 x)$ .

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

Step 3.6

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 $g (x) = 3 x$ .

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Step 3.6.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

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

Step 3.6.2

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

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

Step 3.6.3

Replace all occurrences of $u_{2}$ with $3 x$ .

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

Step 3.7

Combine $\frac{1}{\cos (3 x)}$ and $\sin (3 x)$ .

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

Step 3.8

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 3.9

Multiply $3$ by $- 1$ .

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

Step 3.10

Combine $- 3$ and $\frac{\sin (3 x)}{\cos (3 x)}$ .

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

Step 3.11

Move the negative in front of the fraction.

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

Step 3.12

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

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

Step 3.13

Multiply $- 1$ by $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine factors.

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

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

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

Step 5.3

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{\cos (3 x)}{3 \sin (3 x)}$ .

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

Step 6

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

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

Step 7

Apply L'Hospital's rule.

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Step 7.1

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

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Step 7.1.1

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

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

Step 7.1.2

Evaluate the limit of the numerator.

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Step 7.1.2.1

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

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

Step 7.1.2.2

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

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

Step 7.1.2.3

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

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

Step 7.1.2.4

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

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

Step 7.1.2.5

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

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Step 7.1.2.5.1

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

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

Step 7.1.2.5.2

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

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

Step 7.1.2.6

Simplify the answer.

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Step 7.1.2.6.1

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

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

Step 7.1.2.6.2

Multiply $3$ by $0$ .

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

Step 7.1.2.6.3

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

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

Step 7.1.2.6.4

Multiply $0$ by $1$ .

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

Step 7.1.3

Evaluate the limit of the denominator.

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Step 7.1.3.1

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

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

Step 7.1.3.2

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

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

Step 7.1.3.3

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

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

Step 7.1.3.4

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

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

Step 7.1.3.5

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

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Step 7.1.3.5.1

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

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

Step 7.1.3.5.2

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

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

Step 7.1.3.6

Simplify the answer.

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Step 7.1.3.6.1

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

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

Step 7.1.3.6.2

Multiply $\sin (3 \cdot 0)$ by $1$ .

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

Step 7.1.3.6.3

Multiply $3$ by $0$ .

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

Step 7.1.3.6.4

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

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

Step 7.1.3.6.5

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

Undefined

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

Step 7.1.3.7

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

Undefined

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

Step 7.1.4

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

Undefined

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

Step 7.2

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.

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

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

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

Step 7.3.2

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 $f (x) = \sin (x)$ and $g (x) = \cos (3 x)$ .

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

Step 7.3.3

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 $g (x) = 3 x$ .

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Step 7.3.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 7.3.3.2

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

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

Step 7.3.3.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 7.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 7.3.5

Multiply $3$ by $- 1$ .

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

Step 7.3.6

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

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

Step 7.3.7

Multiply $- 3$ by $1$ .

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

Step 7.3.8

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

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

Step 7.3.9

Reorder terms.

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

Step 7.3.10

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 $f (x) = \cos (x)$ and $g (x) = \sin (3 x)$ .

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

Step 7.3.11

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 $g (x) = 3 x$ .

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Step 7.3.11.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 7.3.11.2

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

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

Step 7.3.11.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 7.3.12

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 7.3.13

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

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

Step 7.3.14

Multiply $3$ by $1$ .

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

Step 7.3.15

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

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

Step 7.3.16

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

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

Step 7.3.17

Reorder terms.

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

Step 8

Evaluate the limit.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

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

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

Step 8.8

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

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

Step 8.9

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

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

Step 8.10

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

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

Step 8.11

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

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

Step 8.12

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

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

Step 8.13

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

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

Step 8.14

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

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

Step 8.15

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

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

Step 8.16

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

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

Step 8.17

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

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

Step 8.18

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

Step 8.19

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

Step 8.20

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

Step 8.21

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

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

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

Step 9.7

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

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

Step 9.8

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

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

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

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

Step 10.1.2

Multiply $- 3$ by $0$ .

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

Step 10.1.3

Multiply $3$ by $0$ .

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

Step 10.1.4

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

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

Step 10.1.5

Multiply $0$ by $0$ .

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

Step 10.1.6

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

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

Step 10.1.7

Multiply $\cos (3 \cdot 0)$ by $1$ .

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

Step 10.1.8

Multiply $3$ by $0$ .

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

Step 10.1.9

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

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

Step 10.1.10

Add $0$ and $1$ .

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

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

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

Step 10.2.2

Multiply $3$ by $1$ .

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

Step 10.2.3

Multiply $3$ by $0$ .

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

Step 10.2.4

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

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

Step 10.2.5

Multiply $3$ by $1$ .

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

Step 10.2.6

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

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

Step 10.2.7

Multiply $- 1$ by $0$ .

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

Step 10.2.8

Multiply $3$ by $0$ .

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

Step 10.2.9

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

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

Step 10.2.10

Multiply $0$ by $0$ .

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

Step 10.2.11

Add $3$ and $0$ .

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

Step 10.3

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

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Step 10.3.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 10.3.2

Multiply $3$ by $3$ .

$\frac{1}{9}$