Calculus Examples

$lim_{x \to 8} \ln (2 x - \ln (x + 1))$

Step 1

Move the limit inside the logarithm.

$\ln (lim_{x \to 8} 2 x - \ln (x + 1))$

Step 2

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

$\ln (lim_{x \to 8} 2 x - lim_{x \to 8} \ln (x + 1))$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\ln (2 lim_{x \to 8} x - lim_{x \to 8} \ln (x + 1))$

Step 4

Move the limit inside the logarithm.

$\ln (2 lim_{x \to 8} x - \ln (lim_{x \to 8} x + 1))$

Step 5

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

$\ln (2 lim_{x \to 8} x - \ln (lim_{x \to 8} x + lim_{x \to 8} 1))$

Step 6

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

$\ln (2 lim_{x \to 8} x - \ln (lim_{x \to 8} x + 1))$

Step 7

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

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Step 7.1

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

$\ln (2 \cdot 8 - \ln (lim_{x \to 8} x + 1))$

Step 7.2

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

$\ln (2 \cdot 8 - \ln (8 + 1))$

Step 8

Simplify each term.

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Step 8.1

Multiply $2$ by $8$ .

$\ln (16 - \ln (8 + 1))$

Step 8.2

Add $8$ and $1$ .

$\ln (16 - \ln (9))$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\ln (16 - \ln (9))$

Decimal Form:

$2.62486968 \dots$