Calculus Examples

$lim_{x \to 8} \frac{\ln (x)}{\sqrt[5]{x}}$

Step 1

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

$\frac{lim_{x \to 8} \ln (x)}{lim_{x \to 8} \sqrt[5]{x}}$

Step 2

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} x)}{lim_{x \to 8} \sqrt[5]{x}}$

Step 3

Move the limit under the radical sign.

$\frac{\ln (lim_{x \to 8} x)}{\sqrt[5]{lim_{x \to 8} x}}$

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 $x$ .

$\frac{\ln (8)}{\sqrt[5]{lim_{x \to 8} x}}$

Step 4.2

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

$\frac{\ln (8)}{\sqrt[5]{8}}$

Step 5

Simplify the answer.

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Step 5.1

Multiply $\frac{\ln (8)}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$\frac{\ln (8)}{\sqrt[5]{8}} \cdot \frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$

Step 5.2

Combine and simplify the denominator.

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Step 5.2.1

Multiply $\frac{\ln (8)}{\sqrt[5]{8}}$ by $\frac{{\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{4}}$ .

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{\sqrt[5]{8} {\sqrt[5]{8}}^{4}}$

Step 5.2.2

Raise $\sqrt[5]{8}$ to the power of $1$ .

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1} {\sqrt[5]{8}}^{4}}$

Step 5.2.3

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

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{1 + 4}}$

Step 5.2.4

Add $1$ and $4$ .

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{{\sqrt[5]{8}}^{5}}$

Step 5.2.5

Rewrite ${\sqrt[5]{8}}^{5}$ as $8$ .

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Step 5.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{8}$ as $8^{\frac{1}{5}}$ .

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{{(8^{\frac{1}{5}})}^{5}}$

Step 5.2.5.2

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

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{8^{\frac{1}{5} \cdot 5}}$

Step 5.2.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 5.2.5.4

Cancel the common factor of $5$ .

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Step 5.2.5.4.1

Cancel the common factor.

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{8^{\frac{5}{5}}}$

Step 5.2.5.4.2

Rewrite the expression.

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{8^{1}}$

Step 5.2.5.5

Evaluate the exponent.

$\frac{\ln (8) {\sqrt[5]{8}}^{4}}{8}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Rewrite ${\sqrt[5]{8}}^{4}$ as $\sqrt[5]{8^{4}}$ .

$\frac{\ln (8) \sqrt[5]{8^{4}}}{8}$

Step 5.3.2

Raise $8$ to the power of $4$ .

$\frac{\ln (8) \sqrt[5]{4096}}{8}$

Step 5.3.3

Rewrite $4096$ as $4^{5} \cdot 4$ .

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Step 5.3.3.1

Factor $1024$ out of $4096$ .

$\frac{\ln (8) \sqrt[5]{1024 (4)}}{8}$

Step 5.3.3.2

Rewrite $1024$ as $4^{5}$ .

$\frac{\ln (8) \sqrt[5]{4^{5} \cdot 4}}{8}$

Step 5.3.4

Pull terms out from under the radical.

$\frac{\ln (8) \cdot 4 \sqrt[5]{4}}{8}$

Step 5.4

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$\frac{\ln (2^{3}) \cdot 4 \sqrt[5]{4}}{8}$

Step 5.5

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \ln (2) \cdot 4 \sqrt[5]{4}}{8}$

Step 5.6

Cancel the common factor of $4$ and $8$ .

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Step 5.6.1

Factor $4$ out of $3 \ln (2) \cdot 4 \sqrt[5]{4}$ .

$\frac{4 (3 \ln (2) \sqrt[5]{4})}{8}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $4$ out of $8$ .

$\frac{4 (3 \ln (2) \sqrt[5]{4})}{4 \cdot 2}$

Step 5.6.2.2

Cancel the common factor.

$\frac{4 (3 \ln (2) \sqrt[5]{4})}{4 \cdot 2}$

Step 5.6.2.3

Rewrite the expression.

$\frac{3 \ln (2) \sqrt[5]{4}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \ln (2) \sqrt[5]{4}}{2}$

Decimal Form:

$1.37191978 \dots$