Calculus Examples

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

Step 1

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

$lim_{n \to 8} {(- 1)}^{n} \cdot lim_{n \to 8} \frac{n}{\ln (n)}$

Step 2

Move the limit into the exponent.

${(- 1)}^{lim_{n \to 8} n} \cdot lim_{n \to 8} \frac{n}{\ln (n)}$

Step 3

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

${(- 1)}^{lim_{n \to 8} n} \cdot \frac{lim_{n \to 8} n}{lim_{n \to 8} \ln (n)}$

Step 4

Move the limit inside the logarithm.

${(- 1)}^{lim_{n \to 8} n} \cdot \frac{lim_{n \to 8} n}{\ln (lim_{n \to 8} n)}$

Step 5

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

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Step 5.1

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

${(- 1)}^{8} \cdot \frac{lim_{n \to 8} n}{\ln (lim_{n \to 8} n)}$

Step 5.2

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

${(- 1)}^{8} \cdot \frac{8}{\ln (lim_{n \to 8} n)}$

Step 5.3

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

${(- 1)}^{8} \cdot \frac{8}{\ln (8)}$

Step 6

Simplify the answer.

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Step 6.1

Raise $- 1$ to the power of $8$ .

$1 \cdot \frac{8}{\ln (8)}$

Step 6.2

Multiply $\frac{8}{\ln (8)}$ by $1$ .

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.84718677 \dots$