Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Since the exponent $x \ln (\frac{1}{x})$ approaches $- \infty$ , the quantity $e^{x \ln (\frac{1}{x})}$ approaches $0$ .

$0$