Calculus Examples

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

Step 1

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

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

Step 2

Move the limit inside the logarithm.

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

Step 3

Move the limit inside the logarithm.

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

Step 4

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

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

Step 5

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

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

Step 6

Move the limit inside the logarithm.

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

Step 7

Move the limit inside the logarithm.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

$\ln (\ln (8 + 1)) - \ln (\ln (8))$

Step 9

Simplify the answer.

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Step 9.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 9.2

Add $8$ and $1$ .

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

Step 10

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.05509564 \dots$