Calculus Examples

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

Step 1

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

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

Step 2

Move the limit inside the logarithm.

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

Step 3

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

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

Step 4

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

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

Step 5

Move the limit into the exponent.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.00004192 \dots$