Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + \sin (x))}^{\csc (x)}$ as $e^{\ln ({(1 + \sin (x))}^{\csc (x)})}$ .

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

Step 1.2

Expand $\ln ({(1 + \sin (x))}^{\csc (x)})$ by moving $\csc (x)$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Move the limit inside the logarithm.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Evaluate $\csc (8)$ .

$e^{7.18529653 \cdot \ln (1 + \sin (8))}$

Step 4.2

Simplify $7.18529653 \cdot \ln (1 + \sin (8))$ by moving $7.18529653$ inside the logarithm.

$e^{\ln ({(1 + \sin (8))}^{7.18529653})}$

Step 4.3

Exponentiation and log are inverse functions.

${(1 + \sin (8))}^{7.18529653}$

Step 4.4

Evaluate $\sin (8)$ .

${(1 + 0.1391731)}^{7.18529653}$

Step 4.5

Add $1$ and $0.1391731$ .

${1.1391731}^{7.18529653}$

Step 4.6

Raise $1.1391731$ to the power of $7.18529653$ .

$2.55043306$