Calculus Examples

$lim_{x \to 0^{+}} {\sin (x)}^{\tan (x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${\sin (x)}^{\tan (x)}$ as $e^{\ln ({\sin (x)}^{\tan (x)})}$ .

$lim_{x \to 0^{+}} e^{\ln ({\sin (x)}^{\tan (x)})}$

Step 1.2

Expand $\ln ({\sin (x)}^{\tan (x)})$ by moving $\tan (x)$ outside the logarithm.

$lim_{x \to 0^{+}} e^{\tan (x) \ln (\sin (x))}$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0^{+}} \tan (x) \ln (\sin (x))}$

Step 3

Rewrite $\tan (x) \ln (\sin (x))$ as $\frac{\ln (\sin (x))}{\frac{1}{\tan (x)}}$ .

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

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.

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

Step 4.1.2

As $x$ approaches $0$ from the right side, $\ln (\sin (x))$ decreases without bound.

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \frac{1}{\tan (x)}}}$

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Apply trigonometric identities.

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Step 4.1.3.1.1

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

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

Step 4.1.3.1.2

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

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \frac{\cos (x)}{\sin (x)}}}$

Step 4.1.3.1.3

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

$e^{\frac{- \infty}{lim_{x \to 0^{+}} \cot (x)}}$

Step 4.1.3.2

As the $x$ values approach $0$ from the right, the function values increase without bound.

$e^{\frac{- \infty}{\infty}}$

Step 4.1.3.3

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{- \infty}{\infty}}$

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{- \infty}{\infty}}$

Step 4.2

Since $\frac{- \infty}{\infty}$ 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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

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 $f (x) = \ln (x)$ and $g (x) = \sin (x)$ .

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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

Step 4.3.2.3

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

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

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

Step 4.3.7

Write $1$ as a fraction with denominator $1$ .

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

Step 4.3.8

Simplify.

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Step 4.3.8.1

Rewrite the expression.

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

Step 4.3.8.2

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

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

Step 4.3.9

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

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

Step 4.3.10

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

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

Step 4.3.11

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

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

Step 4.3.12

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

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

Step 4.3.13

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

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

Step 4.3.14

Add $1$ and $1$ .

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

Step 4.3.15

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

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

Step 4.3.16

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

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

Step 4.3.17

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

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

Step 4.3.18

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

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

Step 4.3.19

Add $1$ and $1$ .

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

Step 4.3.20

Simplify.

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Step 4.3.20.1

Simplify the numerator.

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Step 4.3.20.1.1

Factor $- 1$ out of $- \sin^{2} (x)$ .

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

Step 4.3.20.1.2

Factor $- 1$ out of $- \cos^{2} (x)$ .

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

Step 4.3.20.1.3

Factor $- 1$ out of $- (\sin^{2} (x)) - (\cos^{2} (x))$ .

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

Step 4.3.20.1.4

Apply pythagorean identity.

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

Step 4.3.20.1.5

Multiply $- 1$ by $1$ .

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

Step 4.3.20.2

Move the negative in front of the fraction.

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

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.5

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

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

Step 4.6

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 4.6.1

Factor $\sin (x)$ out of $\cos (x) \sin^{2} (x)$ .

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

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Multiply by $1$ .

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

Step 4.6.2.2

Cancel the common factor.

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

Step 4.6.2.3

Rewrite the expression.

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

Step 4.6.2.4

Divide $\cos (x) \sin (x)$ by $1$ .

$e^{lim_{x \to 0^{+}} - \cos (x) \sin (x)}$

Step 5

Evaluate the limit.

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Step 5.1

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

$e^{- lim_{x \to 0^{+}} \cos (x) \sin (x)}$

Step 5.2

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

$e^{- lim_{x \to 0^{+}} \cos (x) \cdot lim_{x \to 0^{+}} \sin (x)}$

Step 5.3

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

$e^{- \cos (lim_{x \to 0^{+}} x) \cdot lim_{x \to 0^{+}} \sin (x)}$

Step 5.4

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

$e^{- \cos (lim_{x \to 0^{+}} x) \cdot \sin (lim_{x \to 0^{+}} x)}$

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 $x$ .

$e^{- \cos (0) \cdot \sin (lim_{x \to 0^{+}} x)}$

Step 6.2

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

$e^{- \cos (0) \cdot \sin (0)}$

Step 7

Simplify the answer.

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Step 7.1

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

$e^{- 1 \cdot 1 \cdot \sin (0)}$

Step 7.2

Multiply $- 1$ by $1$ .

$e^{- 1 \cdot \sin (0)}$

Step 7.3

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

$e^{- 1 \cdot 0}$

Step 7.4

Multiply $- 1$ by $0$ .

$e^{0}$

Step 8

Anything raised to $0$ is $1$ .

$1$