Calculus Examples

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

Step 1

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

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

Step 2

Move the limit inside the logarithm.

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

Step 3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Move the limit under the radical sign.

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

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the answer.

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Step 7.1

Multiply $5$ by $8$ .

$\frac{\ln (40)}{\sqrt{5 \cdot 8}}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Multiply $5$ by $8$ .

$\frac{\ln (40)}{\sqrt{40}}$

Step 7.2.2

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 7.2.2.1

Factor $4$ out of $40$ .

$\frac{\ln (40)}{\sqrt{4 (10)}}$

Step 7.2.2.2

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

$\frac{\ln (40)}{\sqrt{2^{2} \cdot 10}}$

Step 7.2.3

Pull terms out from under the radical.

$\frac{\ln (40)}{2 \sqrt{10}}$

Step 7.3

Multiply $\frac{\ln (40)}{2 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\ln (40)}{2 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

Multiply $\frac{\ln (40)}{2 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\ln (40) \sqrt{10}}{2 \sqrt{10} \sqrt{10}}$

Step 7.4.2

Move $\sqrt{10}$ .

$\frac{\ln (40) \sqrt{10}}{2 (\sqrt{10} \sqrt{10})}$

Step 7.4.3

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\ln (40) \sqrt{10}}{2 ({\sqrt{10}}^{1} \sqrt{10})}$

Step 7.4.4

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{\ln (40) \sqrt{10}}{2 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})}$

Step 7.4.5

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

$\frac{\ln (40) \sqrt{10}}{2 {\sqrt{10}}^{1 + 1}}$

Step 7.4.6

Add $1$ and $1$ .

$\frac{\ln (40) \sqrt{10}}{2 {\sqrt{10}}^{2}}$

Step 7.4.7

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 7.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{\ln (40) \sqrt{10}}{2 {(10^{\frac{1}{2}})}^{2}}$

Step 7.4.7.2

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

$\frac{\ln (40) \sqrt{10}}{2 \cdot 10^{\frac{1}{2} \cdot 2}}$

Step 7.4.7.3

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

$\frac{\ln (40) \sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}$

Step 7.4.7.4

Cancel the common factor of $2$ .

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Step 7.4.7.4.1

Cancel the common factor.

$\frac{\ln (40) \sqrt{10}}{2 \cdot 10^{\frac{2}{2}}}$

Step 7.4.7.4.2

Rewrite the expression.

$\frac{\ln (40) \sqrt{10}}{2 \cdot 10^{1}}$

Step 7.4.7.5

Evaluate the exponent.

$\frac{\ln (40) \sqrt{10}}{2 \cdot 10}$

Step 7.5

Multiply $2$ by $10$ .

$\frac{\ln (40) \sqrt{10}}{20}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (40) \sqrt{10}}{20}$

Decimal Form:

$0.58326305 \dots$