Calculus Examples

$lim_{x \to 8} \frac{\ln ({(x)}^{2})}{x}$

Step 1

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

$\frac{lim_{x \to 8} \ln ({(x)}^{2})}{lim_{x \to 8} x}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} {(x)}^{2})}{lim_{x \to 8} x}$

Step 3

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

$\frac{\ln ({(lim_{x \to 8} x)}^{2})}{lim_{x \to 8} x}$

Step 4

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

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Step 4.1

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

$\frac{\ln (8^{2})}{lim_{x \to 8} x}$

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Expand $\ln (8^{2})$ by moving $2$ outside the logarithm.

$\frac{2 \ln (8)}{8}$

Step 5.2

Cancel the common factor of $2$ and $8$ .

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Step 5.2.1

Factor $2$ out of $2 \ln (8)$ .

$\frac{2 (\ln (8))}{8}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \ln (8)}{2 \cdot 4}$

Step 5.2.2.2

Cancel the common factor.

$\frac{2 \ln (8)}{2 \cdot 4}$

Step 5.2.2.3

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.51986038 \dots$