Calculus Examples

$lim_{x \to 0} 4 x \ln (\sin (x))$

Step 1

Set up the limit as a left-sided limit.

$lim_{x \to 0^{-}} 4 x \ln (\sin (x))$

Step 2

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

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Step 2.1

Evaluate the limit of $4 x \ln (\sin (x))$ by plugging in $0$ for $x$ .

$4 \cdot 0 \ln (\sin (0))$

Step 2.2

Multiply $4$ by $0$ .

$0 \ln (\sin (0))$

Step 2.3

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

$0 \ln (0)$

Step 2.4

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

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

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

Step 4

Evaluate the right-sided limit.

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Step 4.1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.2

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 4.3

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 $0$ .

$4 \cdot 0$

Step 4.4

Multiply $4$ by $0$ .

$0$

Step 5

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

$Does not exist$