Calculus Examples

$lim_{x \to 0} \frac{\ln (1 + \sin (4 x))}{\frac{1}{\cot (x)}}$

Step 1

Apply trigonometric identities.

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Step 1.1

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

$lim_{x \to 0} \frac{\ln (1 + \sin (4 x))}{\frac{1}{\frac{\cos (x)}{\sin (x)}}}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

$lim_{x \to 0} \frac{\ln (1 + \sin (4 x))}{\frac{\sin (x)}{\cos (x)}}$

Step 1.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$lim_{x \to 0} \frac{\ln (1 + \sin (4 x))}{\tan (x)}$

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.

$\frac{lim_{x \to 0} \ln (1 + \sin (4 x))}{lim_{x \to 0} \tan (x)}$

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.

$\frac{\ln (lim_{x \to 0} 1 + \sin (4 x))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.1.2

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

$\frac{\ln (lim_{x \to 0} 1 + lim_{x \to 0} \sin (4 x))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.1.3

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

$\frac{\ln (1 + lim_{x \to 0} \sin (4 x))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.1.4

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

$\frac{\ln (1 + \sin (lim_{x \to 0} 4 x))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.1.5

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

$\frac{\ln (1 + \sin (4 lim_{x \to 0} x))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.2

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

$\frac{\ln (1 + \sin (4 \cdot 0))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Multiply $4$ by $0$ .

$\frac{\ln (1 + \sin (0))}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.3.1.2

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

$\frac{\ln (1 + 0)}{lim_{x \to 0} \tan (x)}$

Step 2.1.2.3.2

Add $1$ and $0$ .

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

Step 2.1.2.3.3

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

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

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

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

$\frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\ln (1 + \sin (4 x))}{\tan (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + \sin (4 x))]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + \sin (4 x))]}{\frac{d}{d x} [\tan (x)]}$

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 $g (x) = 1 + \sin (4 x)$ .

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Step 2.3.2.1

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

$lim_{x \to 0} \frac{\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [1 + \sin (4 x)]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.2.2

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

$lim_{x \to 0} \frac{\frac{1}{u_{1}} \frac{d}{d x} [1 + \sin (4 x)]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.2.3

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

$lim_{x \to 0} \frac{\frac{1}{1 + \sin (4 x)} \frac{d}{d x} [1 + \sin (4 x)]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.3

By the Sum Rule, the derivative of $1 + \sin (4 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (4 x)]$ .

$lim_{x \to 0} \frac{\frac{1}{1 + \sin (4 x)} (\frac{d}{d x} [1] + \frac{d}{d x} [\sin (4 x)])}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.4

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

$lim_{x \to 0} \frac{\frac{1}{1 + \sin (4 x)} (0 + \frac{d}{d x} [\sin (4 x)])}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.5

Add $0$ and $\frac{d}{d x} [\sin (4 x)]$ .

$lim_{x \to 0} \frac{\frac{1}{1 + \sin (4 x)} \frac{d}{d x} [\sin (4 x)]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.6

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

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Step 2.3.6.1

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

$lim_{x \to 0} \frac{\frac{1}{1 + \sin (4 x)} (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [4 x])}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.6.2

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

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

Step 2.3.6.3

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

$lim_{x \to 0} \frac{\frac{1}{1 + \sin (4 x)} (\cos (4 x) \frac{d}{d x} [4 x])}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.7

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

$lim_{x \to 0} \frac{\frac{\cos (4 x)}{1 + \sin (4 x)} \frac{d}{d x} [4 x]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.8

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

$lim_{x \to 0} \frac{\frac{\cos (4 x)}{1 + \sin (4 x)} (4 \frac{d}{d x} [x])}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.9

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

$lim_{x \to 0} \frac{\frac{4 \cos (4 x)}{1 + \sin (4 x)} \frac{d}{d x} [x]}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.10

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

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

Step 2.3.11

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

$lim_{x \to 0} \frac{\frac{4 \cos (4 x)}{1 + \sin (4 x)}}{\frac{d}{d x} [\tan (x)]}$

Step 2.3.12

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

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

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5

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

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $4$ by $0$ .

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

Step 5.1.2

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

$4 \frac{1}{(1 + \sin (4 \cdot 0)) \cdot \sec^{2} (0)}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $4$ by $0$ .

$4 \frac{1}{(1 + \sin (0)) \sec^{2} (0)}$

Step 5.2.2

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

$4 \frac{1}{(1 + 0) \sec^{2} (0)}$

Step 5.2.3

Add $1$ and $0$ .

$4 \frac{1}{1 \sec^{2} (0)}$

Step 5.2.4

Multiply $\sec^{2} (0)$ by $1$ .

$4 \frac{1}{\sec^{2} (0)}$

Step 5.2.5

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

$4 \frac{1}{1^{2}}$

Step 5.2.6

One to any power is one.

$4 (\frac{1}{1})$

Step 5.3

Cancel the common factor of $1$ .

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Step 5.3.1

Cancel the common factor.

$4 (\frac{1}{1})$

Step 5.3.2

Rewrite the expression.

$4 \cdot 1$

Step 5.4

Multiply $4$ by $1$ .

$4$