Calculus Examples

$lim_{n \to 8} \frac{n \log_{2} (\log_{2} (n))}{{(\ln (n))}^{a}}$

Step 1

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

$\frac{lim_{n \to 8} n \log_{2} (\log_{2} (n))}{lim_{n \to 8} {(\ln (n))}^{a}}$

Step 2

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

$\frac{lim_{n \to 8} n \cdot lim_{n \to 8} \log_{2} (\log_{2} (n))}{lim_{n \to 8} {(\ln (n))}^{a}}$

Step 3

Move the limit inside the logarithm.

$\frac{lim_{n \to 8} n \cdot \log_{2} (lim_{n \to 8} \log_{2} (n))}{lim_{n \to 8} {(\ln (n))}^{a}}$

Step 4

Move the limit inside the logarithm.

$\frac{lim_{n \to 8} n \cdot \log_{2} (\log_{2} (lim_{n \to 8} n))}{lim_{n \to 8} {(\ln (n))}^{a}}$

Step 5

Move the exponent $a$ from ${(\ln (n))}^{a}$ outside the limit using the Limits Power Rule.

$\frac{lim_{n \to 8} n \cdot \log_{2} (\log_{2} (lim_{n \to 8} n))}{{(lim_{n \to 8} \ln (n))}^{a}}$

Step 6

Move the limit inside the logarithm.

$\frac{lim_{n \to 8} n \cdot \log_{2} (\log_{2} (lim_{n \to 8} n))}{{\ln (lim_{n \to 8} n)}^{a}}$

Step 7

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

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Step 7.1

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

$\frac{8 \cdot \log_{2} (\log_{2} (lim_{n \to 8} n))}{{\ln (lim_{n \to 8} n)}^{a}}$

Step 7.2

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

$\frac{8 \cdot \log_{2} (\log_{2} (8))}{{\ln (lim_{n \to 8} n)}^{a}}$

Step 7.3

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

$\frac{8 \cdot \log_{2} (\log_{2} (8))}{{\ln (8)}^{a}}$

Step 8

Logarithm base $2$ of $8$ is $3$ .

$\frac{8 \log_{2} (3)}{{\ln (8)}^{a}}$