Calculus Examples

$lim_{x \to 8} e^{\frac{7}{x} \cdot \ln (e^{x} + 1)}$

Move the limit into the exponent.

$e^{lim_{x \to 8} \frac{7}{x} \cdot \ln (e^{x} + 1)}$

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

$e^{lim_{x \to 8} \frac{7}{x} \cdot lim_{x \to 8} \ln (e^{x} + 1)}$

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$e^{7 lim_{x \to 8} \frac{1}{x} \cdot lim_{x \to 8} \ln (e^{x} + 1)}$

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

$e^{7 \frac{lim_{x \to 8} 1}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (e^{x} + 1)}$

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

$e^{7 \frac{1}{lim_{x \to 8} x} \cdot lim_{x \to 8} \ln (e^{x} + 1)}$

Move the limit inside the logarithm.

$e^{7 \frac{1}{lim_{x \to 8} x} \cdot \ln (lim_{x \to 8} e^{x} + 1)}$

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

$e^{7 \frac{1}{lim_{x \to 8} x} \cdot \ln (lim_{x \to 8} e^{x} + lim_{x \to 8} 1)}$

Move the limit into the exponent.

$e^{7 \frac{1}{lim_{x \to 8} x} \cdot \ln (e^{lim_{x \to 8} x} + lim_{x \to 8} 1)}$

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

$e^{7 \frac{1}{lim_{x \to 8} x} \cdot \ln (e^{lim_{x \to 8} x} + 1)}$

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

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

$e^{7 (\frac{1}{8}) \cdot \ln (e^{lim_{x \to 8} x} + 1)}$

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

$e^{7 (\frac{1}{8}) \cdot \ln (e^{8} + 1)}$

Simplify the answer.

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Combine $7$ and $\frac{1}{8}$ .

$e^{\frac{7}{8} \cdot \ln (e^{8} + 1)}$

Combine $\frac{7}{8}$ and $\ln (e^{8} + 1)$ .

$e^{\frac{7 \ln (e^{8} + 1)}{8}}$

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{7 \ln (e^{8} + 1)}{8}}$

Decimal Form:

$1096.95504619 \dots$