Calculus Examples

$lim_{x \to 0} {(x + \sin (x))}^{\frac{1}{x}}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(x + \sin (x))}^{\frac{1}{x}}$ as $e^{\ln ({(x + \sin (x))}^{\frac{1}{x}})}$ .

$lim_{x \to 0} e^{\ln ({(x + \sin (x))}^{\frac{1}{x}})}$

Step 1.2

Expand $\ln ({(x + \sin (x))}^{\frac{1}{x}})$ by moving $\frac{1}{x}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{1}{x} \ln (x + \sin (x))}$

Step 2

Because there are no values to the left of $0$ in the domain of $e^{\frac{1}{x} \ln (x + \sin (x))}$ , the limit does not exist.

$Does not exist$