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Question Number 58239 by salahahmed last updated on 20/Apr/19

$$\underset{x\to 0^{+}}{\lim }\left(x^{\frac{1}{x}}\right)$$

Answered by salahahmed last updated on 23/Apr/19

$$\underset{x\to 0^{+}}{\lim }\left(e^{\frac{\ln \left(x\right)}{x}}\right)$$$$e^{\underset{x\to 0^{+}}{\lim }\left(\frac{\ln \left(x\right)}{x}\right)}$$$$e^{\underset{x\to 0^{+}}{\lim }\left(\ln \left(x\right)\times \frac{1}{x}\right)}$$$$e^{\ln \left(0^{+}\right)\times \frac{1}{0^{+}}}$$$$e^{\mathrm{\infty }(-\mathrm{\infty })}$$$$e^{-\mathrm{\infty }}=0$$$$\therefore \underset{x\to 0^{+}}{\lim }\left(x^{\frac{1}{x}}\right)=0$$

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