Calculus Examples

$lim_{x \to 8} \ln (x^{2} + 100) - \ln (x)$

Step 1

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

$lim_{x \to 8} \ln (x^{2} + 100) - lim_{x \to 8} \ln (x)$

Step 2

Move the limit inside the logarithm.

$\ln (lim_{x \to 8} x^{2} + 100) - lim_{x \to 8} \ln (x)$

Step 3

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

$\ln (lim_{x \to 8} x^{2} + lim_{x \to 8} 100) - lim_{x \to 8} \ln (x)$

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\ln ({(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 100) - lim_{x \to 8} \ln (x)$

Step 5

Evaluate the limit of $100$ which is constant as $x$ approaches $8$ .

$\ln ({(lim_{x \to 8} x)}^{2} + 100) - lim_{x \to 8} \ln (x)$

Step 6

Move the limit inside the logarithm.

$\ln ({(lim_{x \to 8} x)}^{2} + 100) - \ln (lim_{x \to 8} x)$

Step 7

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

Tap for more steps...

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

$\ln (8^{2} + 100) - \ln (lim_{x \to 8} x)$

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

$\ln (8^{2} + 100) - \ln (8)$

Step 8

Simplify the answer.

Tap for more steps...

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

$\ln (\frac{8^{2} + 100}{8})$

Simplify the numerator.

Tap for more steps...

Raise $8$ to the power of $2$ .

$\ln (\frac{64 + 100}{8})$

Add $64$ and $100$ .

$\ln (\frac{164}{8})$

Cancel the common factor of $164$ and $8$ .

Tap for more steps...

Factor $4$ out of $164$ .

$\ln (\frac{4 (41)}{8})$

Cancel the common factors.

Tap for more steps...

Factor $4$ out of $8$ .

$\ln (\frac{4 \cdot 41}{4 \cdot 2})$

Cancel the common factor.

$\ln (\frac{4 \cdot 41}{4 \cdot 2})$

Rewrite the expression.

$\ln (\frac{41}{2})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{41}{2})$

Decimal Form:

$3.02042488 \dots$