Calculus Examples

$lim_{x \to 0^{+}} 12 {\sin (x)}^{14 x}$

Step 1

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$12 lim_{x \to 0^{+}} {\sin (x)}^{14 x}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite ${\sin (x)}^{14 x}$ as $e^{\ln ({\sin (x)}^{14 x})}$ .

$12 lim_{x \to 0^{+}} e^{\ln ({\sin (x)}^{14 x})}$

Step 2.2

Expand $\ln ({\sin (x)}^{14 x})$ by moving $14 x$ outside the logarithm.

$12 lim_{x \to 0^{+}} e^{14 x \ln (\sin (x))}$

Step 3

Evaluate the limit.

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Step 3.1

Move the limit into the exponent.

$12 e^{lim_{x \to 0^{+}} 14 x \ln (\sin (x))}$

Step 3.2

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$12 e^{14 lim_{x \to 0^{+}} x \ln (\sin (x))}$

Step 4

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

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 $0$ .

$12 e^{14 \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Multiply $14$ by $0$ .

$12 e^{0}$

Step 6.2

Anything raised to $0$ is $1$ .

$12 \cdot 1$

Step 6.3

Multiply $12$ by $1$ .

$12$