Calculus Examples

$lim_{x \to 8} \frac{1}{x^{\sin (x) + 2}}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{\sin (x) + 2}}$

Step 1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{1}{lim_{x \to 8} x^{\sin (x) + 2}}$

Step 2

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 2.1

Rewrite $x^{\sin (x) + 2}$ as $e^{\ln (x^{\sin (x) + 2})}$ .

$\frac{1}{lim_{x \to 8} e^{\ln (x^{\sin (x) + 2})}}$

Step 2.2

Expand $\ln (x^{\sin (x) + 2})$ by moving $\sin (x) + 2$ outside the logarithm.

$\frac{1}{lim_{x \to 8} e^{(\sin (x) + 2) \ln (x)}}$

Step 3

Evaluate the limit.

Tap for more steps...

Step 3.1

Move the limit into the exponent.

$\frac{1}{e^{lim_{x \to 8} (\sin (x) + 2) \ln (x)}}$

Step 3.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{1}{e^{lim_{x \to 8} \sin (x) + 2 \cdot lim_{x \to 8} \ln (x)}}$

Step 3.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{1}{e^{(lim_{x \to 8} \sin (x) + lim_{x \to 8} 2) \cdot lim_{x \to 8} \ln (x)}}$

Step 3.4

Move the limit inside the trig function because sine is continuous.

$\frac{1}{e^{(\sin (lim_{x \to 8} x) + lim_{x \to 8} 2) \cdot lim_{x \to 8} \ln (x)}}$

Step 3.5

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{1}{e^{(\sin (lim_{x \to 8} x) + 2) \cdot lim_{x \to 8} \ln (x)}}$

Step 3.6

Move the limit inside the logarithm.

$\frac{1}{e^{(\sin (lim_{x \to 8} x) + 2) \cdot \ln (lim_{x \to 8} x)}}$

Step 4

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

Tap for more steps...

Step 4.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1}{e^{(\sin (8) + 2) \cdot \ln (lim_{x \to 8} x)}}$

Step 4.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1}{e^{(\sin (8) + 2) \cdot \ln (8)}}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Simplify the denominator.

Tap for more steps...

Step 5.1.1

Evaluate $\sin (8)$ .

$\frac{1}{e^{(0.1391731 + 2) \cdot \ln (8)}}$

Step 5.1.2

Add $0.1391731$ and $2$ .

$\frac{1}{e^{2.1391731 \cdot \ln (8)}}$

Step 5.1.3

Simplify $2.1391731 \cdot \ln (8)$ by moving $2.1391731$ inside the logarithm.

$\frac{1}{e^{\ln (8^{2.1391731})}}$

Step 5.1.4

Exponentiation and log are inverse functions.

$\frac{1}{8^{2.1391731}}$

Step 5.1.5

Raise $8$ to the power of $2.1391731$ .

$\frac{1}{85.48025477}$

Step 5.2

Divide $1$ by $85.48025477$ .

$0.0116986$