Evaluate limit as x approaches 8 of ( natural log of 3x)/( square root of 3x)

$lim_{x \to 8} \frac{\ln (3 x)}{\sqrt{3 x}}$

Take the limit of each term.

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

$\frac{(lim_{x \to 8} \ln (3 x))}{(lim_{x \to 8} \sqrt{3 x})}$

Move the limit inside the logarithm.

$\frac{\ln (lim_{x \to 8} 3 x)}{lim_{x \to 8} \sqrt{3 x}}$

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (3 lim_{x \to 8} x)}{lim_{x \to 8} \sqrt{3 x}}$

Move the limit under the radical sign.

$\frac{\ln (3 lim_{x \to 8} x)}{\sqrt{lim_{x \to 8} 3 x}}$

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\ln (3 lim_{x \to 8} x)}{\sqrt{3 lim_{x \to 8} x}}$

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

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

$\frac{\ln (3 \cdot 8)}{\sqrt{3 lim_{x \to 8} x}}$

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

$\frac{\ln (3 \cdot 8)}{\sqrt{3 \cdot 8}}$

Simplify the answer.

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Multiply $3$ by $8$ .

$\frac{\ln (24)}{\sqrt{3 \cdot 8}}$

Simplify the denominator.

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Multiply $3$ by $8$ .

$\frac{\ln (24)}{\sqrt{24}}$

Rewrite $24$ as $2^{2} \cdot 6$ .

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Factor $4$ out of $24$ .

$\frac{\ln (24)}{\sqrt{4 (6)}}$

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

$\frac{\ln (24)}{\sqrt{2^{2} \cdot 6}}$

Pull terms out from under the radical.

$\frac{\ln (24)}{2 \sqrt{6}}$

Multiply $\frac{\ln (24)}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\ln (24)}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Combine and simplify the denominator.

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Multiply $\frac{\ln (24)}{2 \sqrt{6}}$ and $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\ln (24) \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Move $\sqrt{6}$ .

$\frac{\ln (24) \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\ln (24) \sqrt{6}}{2 ({\sqrt{6}}^{1} \sqrt{6})}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\ln (24) \sqrt{6}}{2 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

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

$\frac{\ln (24) \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\ln (24) \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\ln (24) \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

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

$\frac{\ln (24) \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

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

$\frac{\ln (24) \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{\ln (24) \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\frac{\ln (24) \sqrt{6}}{2 \cdot 6^{1}}$

Evaluate the exponent.

$\frac{\ln (24) \sqrt{6}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$\frac{\ln (24) \sqrt{6}}{12}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (24) \sqrt{6}}{12}$

Decimal Form:

$0.64871752 \dots$

Evaluate limit as x approaches 8 of ( natural log of 3x)/( square root of 3x)

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