Calculus Examples

$lim_{x \to 0} 1 + {\sin (4 x)}^{\cot (x)}$

Step 1

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} 1 + {\sin (4 x)}^{\cot (x)}$

Step 2

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

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Step 2.1

Evaluate the limit of $1 + {\sin (4 x)}^{\cot (x)}$ by plugging in $0$ for $x$ .

$1 + {\sin (4 \cdot 0)}^{\cot (0)}$

Step 2.2

Rewrite $\cot (0)$ in terms of sines and cosines.

$1 + {\sin (4 \cdot 0)}^{\frac{\cos (0)}{\sin (0)}}$

Step 2.3

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

$1 + {\sin (4 \cdot 0)}^{\frac{\cos (0)}{0}}$

Step 2.4

Since $1 + {\sin (4 \cdot 0)}^{\frac{\cos (0)}{0}}$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

$lim_{x \to 0^{+}} 1 + {\sin (4 x)}^{\cot (x)}$

Step 4

Evaluate the right-sided limit.

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Step 4.1

Evaluate the limit.

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Step 4.1.1

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

$lim_{x \to 0^{+}} 1 + lim_{x \to 0^{+}} {\sin (4 x)}^{\cot (x)}$

Step 4.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

$1 + lim_{x \to 0^{+}} {\sin (4 x)}^{\cot (x)}$

Step 4.2

Use the properties of logarithms to simplify the limit.

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Step 4.2.1

Rewrite ${\sin (4 x)}^{\cot (x)}$ as $e^{\ln ({\sin (4 x)}^{\cot (x)})}$ .

$1 + lim_{x \to 0^{+}} e^{\ln ({\sin (4 x)}^{\cot (x)})}$

Step 4.2.2

Expand $\ln ({\sin (4 x)}^{\cot (x)})$ by moving $\cot (x)$ outside the logarithm.

$1 + lim_{x \to 0^{+}} e^{\cot (x) \ln (\sin (4 x))}$

Step 4.3

Since the exponent $\cot (x) \ln (\sin (4 x))$ approaches $- \infty$ , the quantity $e^{\cot (x) \ln (\sin (4 x))}$ approaches $0$ .

$1 + 0$

Step 4.4

Add $1$ and $0$ .

$1$

Step 5

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

$Does not exist$