Calculus Examples

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

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

Step 1

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 1.1

Rewrite

{tan (x)}^{cos (x)}

${\tan (x)}^{\cos (x)}$ as

e^{ln ({tan (x)}^{cos (x)})}

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

lim x \to \frac{π}{2} 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)})

$\ln ({\tan (x)}^{\cos (x)})$ by moving

cos (x)

$\cos (x)$ outside the logarithm.

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

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

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

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

Step 2

Set up the limit as a left-sided limit.

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

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

Step 3

Evaluate the left-sided limit.

Tap for more steps...

Step 3.1

Move the limit into the exponent.

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

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

Step 3.2

Rewrite

cos (x) ln (tan (x))

$\cos (x) \ln (\tan (x))$ as

\frac{ln (tan (x))}{\frac{1}{cos (x)}}

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

e^{lim x \to {(\frac{π}{2})}^{-} \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.

Tap for more steps...

Step 3.3.1

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

Tap for more steps...

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)}}}

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

Step 3.3.1.2

As log approaches infinity, the value goes to

\infty

$\infty$ .

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

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

Step 3.3.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 3.3.1.3.1

Convert from

\frac{1}{cos (x)}

$\frac{1}{\cos (x)}$ to

sec (x)

$\sec (x)$ .

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

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

Step 3.3.1.3.2

As the

x

$x$ values approach

\frac{π}{2}

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

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

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

Step 3.3.1.3.3

Infinity divided by infinity is undefined.

Undefined

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

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

Step 3.3.1.4

Infinity divided by infinity is undefined.

Undefined

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

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

Step 3.3.2

Since

\frac{\infty}{\infty}

$\frac{\infty}{\infty}$ 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)}]}

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

Step 3.3.3

Find the derivative of the numerator and denominator.

Tap for more steps...

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)}]}}

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

Step 3.3.3.2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \ln (x)$ and

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

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 3.3.3.9.1

Simplify the numerator.

Tap for more steps...

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

$\cos (x)$ .

Tap for more steps...

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.

Tap for more steps...

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

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

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

$\cos (x)$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

$x$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

$Does not exist$