Calculus Examples

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

Step 1

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

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

Step 2

Move the limit inside the logarithm.

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

Step 3

Move the limit under the radical sign.

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

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 $x$ .

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

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

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Step 5.1.1.1

Factor $4$ out of $8$ .

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

Step 5.1.1.2

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

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

Step 5.1.2

Pull terms out from under the radical.

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

Step 5.2

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

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

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

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

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

Step 5.3.2

Move $\sqrt{2}$ .

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

Step 5.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.6

Add $1$ and $1$ .

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

Step 5.3.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 5.3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.3.7.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 5.3.7.4

Cancel the common factor of $2$ .

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Step 5.3.7.4.1

Cancel the common factor.

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

Step 5.3.7.4.2

Rewrite the expression.

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

Step 5.3.7.5

Evaluate the exponent.

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

Step 5.4

Multiply $2$ by $2$ .

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.73519360 \dots$