Calculus Examples

$lim_{x \to 0^{+}} {\sinh (3 x)}^{x}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${\sinh (3 x)}^{x}$ as $e^{\ln ({\sinh (3 x)}^{x})}$ .

$lim_{x \to 0^{+}} e^{\ln ({\sinh (3 x)}^{x})}$

Step 1.2

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

$lim_{x \to 0^{+}} e^{x \ln (\sinh (3 x))}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 0^{+}} x \ln (\sinh (3 x))}$

Step 2.2

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

$e^{lim_{x \to 0^{+}} x \cdot lim_{x \to 0^{+}} \ln (\sinh (3 x))}$

Step 2.3

Move the limit inside the logarithm.

$e^{lim_{x \to 0^{+}} x \cdot \ln (lim_{x \to 0^{+}} \sinh (3 x))}$

Step 3

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

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Step 3.1

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

$e^{0 \cdot \ln (lim_{x \to 0^{+}} \sinh (3 x))}$

Step 3.2

Evaluate the limit of $\sinh (3 x)$ by plugging in $0$ for $x$ .

$e^{0 \cdot \ln (\sinh (3 \cdot 0))}$

Step 3.3

Multiply $3$ by $0$ .

$e^{0 \cdot \ln (\sinh (0))}$

Step 4

Simplify the answer.

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Step 4.1

Simplify $0 \cdot \ln (\sinh (0))$ by moving $0$ inside the logarithm.

$e^{\ln (\sinh^{0} (0))}$

Step 4.2

Exponentiation and log are inverse functions.

$\sinh^{0} (0)$

Step 4.3

Anything raised to $0$ is $1$ .

$1$