Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Move the limit into the exponent.

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

Step 3

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

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

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.

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

Step 4.1.2

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

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = \sin (x)$ .

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Step 4.3.4.1

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

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

Step 4.3.4.2

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

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

Step 4.3.4.3

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

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

Step 4.3.5

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

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

Step 4.3.6

Simplify.

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Step 4.3.6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3.6.2

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

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

Step 4.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

Separate fractions.

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

Step 4.8

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

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

Step 4.9

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

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

Step 5

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

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

Step 6

Apply L'Hospital's rule.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Evaluate the limit of the numerator.

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

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

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

Step 6.1.2.4

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

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Step 6.1.2.4.1

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

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

Step 6.1.2.4.2

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

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

Step 6.1.2.5

Simplify the answer.

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Step 6.1.2.5.1

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

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

Step 6.1.2.5.2

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

$e^{- \frac{0 \cdot 0}{lim_{x \to 0^{+}} x}}$

Step 6.1.2.5.3

Multiply $0$ by $0$ .

$e^{- \frac{0}{lim_{x \to 0^{+}} x}}$

Step 6.1.3

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

$e^{- \frac{0}{0}}$

Step 6.1.4

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

Undefined

$e^{- \frac{0}{0}}$

Step 6.2

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.

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

Step 6.3

Find the derivative of the numerator and denominator.

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Step 6.3.1

Differentiate the numerator and denominator.

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.3.4

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

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

Step 6.3.5

Simplify.

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Step 6.3.5.1

Reorder terms.

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

Step 6.3.5.2

Simplify each term.

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Step 6.3.5.2.1

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

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

Step 6.3.5.2.2

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

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

Step 6.3.5.2.3

One to any power is one.

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

Step 6.3.5.2.4

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

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

Step 6.3.5.2.5

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 6.3.5.2.5.1

Reorder $\cos (x)$ and $\tan (x)$ .

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

Step 6.3.5.2.5.2

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

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

Step 6.3.5.2.5.3

Cancel the common factors.

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

Step 6.3.6

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

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

Step 6.4

Combine terms.

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Step 6.4.1

To write $\sin (x)$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

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

Step 6.4.2

Combine the numerators over the common denominator.

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

Step 6.5

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

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

Step 7

Evaluate the limit.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

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

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

Step 7.9

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

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

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

Step 9.1.2

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

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

Step 9.1.3

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

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

Step 9.1.4

One to any power is one.

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

Step 9.1.5

Multiply $0$ by $1$ .

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

Step 9.1.6

Add $0$ and $0$ .

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

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

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

Step 9.2.2

One to any power is one.

$e^{- \frac{0}{1}}$

Step 9.3

Divide $0$ by $1$ .

$e^{- 0}$

Step 9.4

Multiply $- 1$ by $0$ .

$e^{0}$

Step 10

Anything raised to $0$ is $1$ .

$1$