Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Because there is a gap in the domain of $e^{x \ln (\frac{x}{x + 1})}$ to the left of $0$ , the limit does not exist.

$Does not exist$