Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(e^{x} + 1)}^{\frac{2}{x}}$ as $e^{\ln ({(e^{x} + 1)}^{\frac{2}{x}})}$ .

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

Step 1.2

Expand $\ln ({(e^{x} + 1)}^{\frac{2}{x}})$ by moving $\frac{2}{x}$ outside the logarithm.

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

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

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

Step 2.3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Move the limit inside the logarithm.

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

Step 2.7

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

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

Step 2.8

Move the limit into the exponent.

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

Step 2.9

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

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

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 $x$ .

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Cancel the common factor of $2$ .

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Step 4.1.1

Factor $2$ out of $8$ .

$e^{2 \frac{1}{2 (4)} \cdot \ln (e^{8} + 1)}$

Step 4.1.2

Cancel the common factor.

$e^{2 \frac{1}{2 \cdot 4} \cdot \ln (e^{8} + 1)}$

Step 4.1.3

Rewrite the expression.

$e^{\frac{1}{4} \cdot \ln (e^{8} + 1)}$

Step 4.2

Combine $\frac{1}{4}$ and $\ln (e^{8} + 1)$ .

$e^{\frac{\ln (e^{8} + 1)}{4}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{\ln (e^{8} + 1)}{4}}$

Decimal Form:

$7.38967570 \dots$