Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

$0$