Calculus Examples

$lim_{x \to 8} \frac{\ln (x^{\frac{1}{2}})}{\ln (x) \ln (x)}$

Step 1

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

$\frac{lim_{x \to 8} \ln (x^{\frac{1}{2}})}{lim_{x \to 8} \ln (x) \ln (x)}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} x^{\frac{1}{2}})}{lim_{x \to 8} \ln (x) \ln (x)}$

Step 3

Move the exponent $\frac{1}{2}$ from $x^{\frac{1}{2}}$ outside the limit using the Limits Power Rule.

$\frac{\ln ({(lim_{x \to 8} x)}^{\frac{1}{2}})}{lim_{x \to 8} \ln (x) \ln (x)}$

Step 4

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

$\frac{\ln ({(lim_{x \to 8} x)}^{\frac{1}{2}})}{lim_{x \to 8} \ln (x) \cdot lim_{x \to 8} \ln (x)}$

Step 5

Move the limit inside the logarithm.

$\frac{\ln ({(lim_{x \to 8} x)}^{\frac{1}{2}})}{\ln (lim_{x \to 8} x) \cdot lim_{x \to 8} \ln (x)}$

Step 6

Move the limit inside the logarithm.

$\frac{\ln ({(lim_{x \to 8} x)}^{\frac{1}{2}})}{\ln (lim_{x \to 8} x) \cdot \ln (lim_{x \to 8} x)}$

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 $x$ .

$\frac{\ln (8^{\frac{1}{2}})}{\ln (lim_{x \to 8} x) \cdot \ln (lim_{x \to 8} x)}$

Step 7.2

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

$\frac{\ln (8^{\frac{1}{2}})}{\ln (8) \cdot \ln (lim_{x \to 8} x)}$

Step 7.3

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

$\frac{\ln (8^{\frac{1}{2}})}{\ln (8) \cdot \ln (8)}$

Step 8

Simplify the answer.

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Step 8.1

Expand $\ln (8^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{\frac{1}{2} \ln (8)}{\ln (8) \cdot \ln (8)}$

Step 8.2

Cancel the common factor of $\ln (8)$ .

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Step 8.2.1

Cancel the common factor.

$\frac{\frac{1}{2} \ln (8)}{\ln (8) \cdot \ln (8)}$

Step 8.2.2

Rewrite the expression.

$\frac{\frac{1}{2}}{\ln (8)}$

Step 8.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{1}{\ln (8)}$

Step 8.4

Multiply $\frac{1}{2}$ by $\frac{1}{\ln (8)}$ .

$\frac{1}{2 \ln (8)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2 \ln (8)}$

Decimal Form:

$0.24044917 \dots$