Calculus Examples

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

Step 1

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

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

Step 2

Move the limit inside the logarithm.

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

Step 3

Move the limit under the radical sign.

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1.1.1

Factor $4$ out of $8$ .

$\frac{\ln (\sqrt{4 (2)})}{8^{2}}$

Step 6.1.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 6.1.2

Pull terms out from under the radical.

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

Step 6.2

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

$\frac{\ln (2 \sqrt{2})}{64}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (2 \sqrt{2})}{64}$

Decimal Form:

$0.01624563 \dots$