Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Set up the limit as a left-sided limit.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

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

Step 5

Evaluate the right-sided limit.

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Step 5.1

Move the limit into the exponent.

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

Step 5.2

Make a table to show the behavior of the function $x \ln (\sin (x))$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & x \ln (\sin (x)) \\ 0.1 & - 0.23042523 \\ 0.01 & - 0.04605186 \\ 0.001 & - 0.00690775 \\ 0.0001 & - 0.00092103 \\ 0.00001 & - 0.00011512 \\ 0.000001 & - 0.00001381 \end{array}$

Step 5.3

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $x \ln (\sin (x))$ as $x$ approaches $0$ from the right is $0$ .

$e^{0}$

Step 5.4

Anything raised to $0$ is $1$ .

$1$

Step 6

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

$Does not exist$