Calculus Examples

$lim_{x \to \frac{π}{2}} {\tan (x)}^{\cos (x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

$lim_{x \to \frac{π}{2}} e^{\ln ({\tan (x)}^{\cos (x)})}$

Step 1.2

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

$lim_{x \to \frac{π}{2}} e^{\cos (x) \ln (\tan (x))}$

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to {(\frac{π}{2})}^{-}} e^{\cos (x) \ln (\tan (x))}$

Step 3

Evaluate the left-sided limit.

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Step 3.1

Move the limit into the exponent.

$e^{lim_{x \to {(\frac{π}{2})}^{-}} \cos (x) \ln (\tan (x))}$

Step 3.2

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

$e^{lim_{x \to {(\frac{π}{2})}^{-}} \frac{\ln (\tan (x))}{\frac{1}{\cos (x)}}}$

Step 3.3

Apply L'Hospital's rule.

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Step 3.3.1

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

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Step 3.3.1.1

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

$e^{\frac{lim_{x \to {(\frac{π}{2})}^{-}} \ln (\tan (x))}{lim_{x \to {(\frac{π}{2})}^{-}} \frac{1}{\cos (x)}}}$

Step 3.3.1.2

As log approaches infinity, the value goes to $\infty$ .

$e^{\frac{\infty}{lim_{x \to {(\frac{π}{2})}^{-}} \frac{1}{\cos (x)}}}$

Step 3.3.1.3

Evaluate the limit of the denominator.

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Step 3.3.1.3.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$e^{\frac{\infty}{lim_{x \to {(\frac{π}{2})}^{-}} \sec (x)}}$

Step 3.3.1.3.2

As the $x$ values approach $\frac{π}{2}$ from the left, the function values increase without bound.

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

Step 3.3.1.3.3

Infinity divided by infinity is undefined.

Undefined

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

Step 3.3.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 3.3.2

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.

$lim_{x \to {(\frac{π}{2})}^{-}} \frac{\ln (\tan (x))}{\frac{1}{\cos (x)}} = lim_{x \to {(\frac{π}{2})}^{-}} \frac{\frac{d}{d x} [\ln (\tan (x))]}{\frac{d}{d x} [\frac{1}{\cos (x)}]}$

Step 3.3.3

Find the derivative of the numerator and denominator.

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Step 3.3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.3.2

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

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Step 3.3.3.2.1

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

$e^{lim_{x \to {(\frac{π}{2})}^{-}} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [\frac{1}{\cos (x)}]}}$

Step 3.3.3.2.2

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

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

Step 3.3.3.2.3

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

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

Step 3.3.3.3

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

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

Step 3.3.3.4

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

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

Step 3.3.3.5

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

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

Step 3.3.3.6

Simplify.

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Step 3.3.3.6.1

Rewrite the expression.

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

Step 3.3.3.6.2

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

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

Step 3.3.3.7

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

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

Step 3.3.3.8

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

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

Step 3.3.3.9

Simplify.

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Step 3.3.3.9.1

Simplify the numerator.

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Step 3.3.3.9.1.1

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

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

Step 3.3.3.9.1.2

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

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

Step 3.3.3.9.1.3

Cancel the common factor of $\cos (x)$ .

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Step 3.3.3.9.1.3.1

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

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

Step 3.3.3.9.1.3.2

Cancel the common factor.

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

Step 3.3.3.9.1.3.3

Rewrite the expression.

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

Step 3.3.3.9.1.4

One to any power is one.

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

Step 3.3.3.9.2

Combine terms.

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Step 3.3.3.9.2.1

Rewrite $\frac{\frac{1}{\cos (x)}}{\sin (x)}$ as a product.

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

Step 3.3.3.9.2.2

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

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

Step 3.3.3.10

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

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

Step 3.3.3.11

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

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Step 3.3.3.11.1

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

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

Step 3.3.3.11.2

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

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

Step 3.3.3.11.3

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

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

Step 3.3.3.12

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

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

Step 3.3.3.13

Multiply $- 1$ by $- 1$ .

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

Step 3.3.3.14

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

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

Step 3.3.3.15

Simplify.

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Step 3.3.3.15.1

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

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

Step 3.3.3.15.2

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

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

Step 3.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.5

Combine factors.

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Step 3.3.5.1

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

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

Step 3.3.5.2

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

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

Step 3.3.5.3

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

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

Step 3.3.5.4

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

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

Step 3.3.5.5

Add $1$ and $1$ .

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

Step 3.3.6

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

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Step 3.3.6.1

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

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

Step 3.3.6.2

Cancel the common factors.

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Step 3.3.6.2.1

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

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

Step 3.3.6.2.2

Cancel the common factor.

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

Step 3.3.6.2.3

Rewrite the expression.

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

Step 3.3.7

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

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

Step 3.3.8

Separate fractions.

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

Step 3.3.9

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

$e^{lim_{x \to {(\frac{π}{2})}^{-}} \frac{1}{\sin (x)} \cot (x)}$

Step 3.3.10

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

$e^{lim_{x \to {(\frac{π}{2})}^{-}} \csc (x) \cot (x)}$

Step 3.4

Evaluate the limit.

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Step 3.4.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$e^{lim_{x \to {(\frac{π}{2})}^{-}} \csc (x) \cdot lim_{x \to {(\frac{π}{2})}^{-}} \cot (x)}$

Step 3.4.2

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

$e^{\csc (lim_{x \to {(\frac{π}{2})}^{-}} x) \cdot lim_{x \to {(\frac{π}{2})}^{-}} \cot (x)}$

Step 3.4.3

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

$e^{\csc (lim_{x \to {(\frac{π}{2})}^{-}} x) \cdot \cot (lim_{x \to {(\frac{π}{2})}^{-}} x)}$

Step 3.5

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 3.5.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$e^{\csc (\frac{π}{2}) \cdot \cot (lim_{x \to {(\frac{π}{2})}^{-}} x)}$

Step 3.5.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$e^{\csc (\frac{π}{2}) \cdot \cot (\frac{π}{2})}$

Step 3.6

Simplify the answer.

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Step 3.6.1

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$e^{1 \cdot \cot (\frac{π}{2})}$

Step 3.6.2

Multiply $\cot (\frac{π}{2})$ by $1$ .

$e^{\cot (\frac{π}{2})}$

Step 3.6.3

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$e^{0}$

Step 3.7

Anything raised to $0$ is $1$ .

$1$

Step 4

Set up the limit as a right-sided limit.

$lim_{x \to {(\frac{π}{2})}^{+}} e^{\cos (x) \ln (\tan (x))}$

Step 5

Evaluate the limits by plugging in the value for the variable.

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Step 5.1

Evaluate the limit of $e^{\cos (x) \ln (\tan (x))}$ by plugging in $\frac{π}{2}$ for $x$ .

$e^{\cos (\frac{π}{2}) \ln (\tan (\frac{π}{2}))}$

Step 5.2

Rewrite $\tan (\frac{π}{2})$ in terms of sines and cosines.

$e^{\cos (\frac{π}{2}) \ln (\frac{\sin (\frac{π}{2})}{\cos (\frac{π}{2})})}$

Step 5.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$e^{\cos (\frac{π}{2}) \ln (\frac{\sin (\frac{π}{2})}{0})}$

Step 5.4

Since $e^{\cos (\frac{π}{2}) \ln (\frac{\sin (\frac{π}{2})}{0})}$ is undefined, the limit does not exist.

$Does not exist$

Step 6

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$