Calculus Examples

$lim_{x \to 0} \sqrt{\frac{3^{x}}{\log (x + 10)}}$

Step 1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 0} \frac{3^{x}}{\log (x + 10)}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\sqrt{\frac{lim_{x \to 0} 3^{x}}{lim_{x \to 0} \log (x + 10)}}$

Step 3

Move the limit into the exponent.

$\sqrt{\frac{3^{lim_{x \to 0} x}}{lim_{x \to 0} \log (x + 10)}}$

Step 4

Move the limit inside the logarithm.

$\sqrt{\frac{3^{lim_{x \to 0} x}}{\log (lim_{x \to 0} x + 10)}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\sqrt{\frac{3^{lim_{x \to 0} x}}{\log (lim_{x \to 0} x + lim_{x \to 0} 10)}}$

Step 6

Evaluate the limit of $10$ which is constant as $x$ approaches $0$ .

$\sqrt{\frac{3^{lim_{x \to 0} x}}{\log (lim_{x \to 0} x + 10)}}$

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt{\frac{3^{0}}{\log (lim_{x \to 0} x + 10)}}$

Step 7.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\sqrt{\frac{3^{0}}{\log (0 + 10)}}$

Step 8

Simplify the answer.

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Step 8.1

Anything raised to $0$ is $1$ .

$\sqrt{\frac{1}{\log (0 + 10)}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Add $0$ and $10$ .

$\sqrt{\frac{1}{\log (10)}}$

Step 8.2.2

Logarithm base $10$ of $10$ is $1$ .

$\sqrt{\frac{1}{1}}$

Step 8.3

Divide $1$ by $1$ .

$\sqrt{1}$

Step 8.4

Any root of $1$ is $1$ .

$1$