Calculus Examples

$\frac{lim_{n \to 8} \ln (1 + \frac{1}{n})}{n}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit inside the logarithm.

$\frac{\ln (lim_{n \to 8} 1 + \frac{1}{n})}{n}$

Step 1.2

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

$\frac{\ln (lim_{n \to 8} 1 + lim_{n \to 8} \frac{1}{n})}{n}$

Step 1.3

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

$\frac{\ln (1 + lim_{n \to 8} \frac{1}{n})}{n}$

Step 1.4

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

$\frac{\ln (1 + \frac{lim_{n \to 8} 1}{lim_{n \to 8} n})}{n}$

Step 1.5

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

$\frac{\ln (1 + \frac{1}{lim_{n \to 8} n})}{n}$

Step 2

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

$\frac{\ln (1 + \frac{1}{8})}{n}$

Step 3

Simplify the numerator.

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Step 3.1

Write $1$ as a fraction with a common denominator.

$\frac{\ln (\frac{8}{8} + \frac{1}{8})}{n}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\ln (\frac{8 + 1}{8})}{n}$

Step 3.3

Add $8$ and $1$ .

$\frac{\ln (\frac{9}{8})}{n}$