Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} e^{\cos (x) \ln (\tan (x))}$

Step 3

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

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Step 3.1

Evaluate the limit of $e^{\cos (x) \ln (\tan (x))}$ by plugging in $0$ for $x$ .

$e^{\cos (0) \ln (\tan (0))}$

Step 3.2

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

$e^{1 \ln (\tan (0))}$

Step 3.3

Multiply $\ln (\tan (0))$ by $1$ .

$e^{\ln (\tan (0))}$

Step 3.4

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

$e^{\ln (0)}$

Step 3.5

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

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

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

Step 5

Since the exponent $\cos (x) \ln (\tan (x))$ approaches $- \infty$ , the quantity $e^{\cos (x) \ln (\tan (x))}$ approaches $0$ .

$0$

Step 6

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

$Does not exist$