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Question Number 1766 by Rasheed Ahmad last updated on 18/Sep/15

$$Determine$$$$\left(i\right)\underset{a\to \mathrm{\infty }}{lim}{a}^{\frac{1}{a}}\left(ii\right)\underset{a\to 0}{lim}{a}^{\frac{1}{a}}$$

Answered by 123456 last updated on 19/Sep/15

$$\mathrm{L}=\underset{x\to +\mathrm{\infty }}{\lim }{x}^{\frac{1}{x}}(\to {\mathrm{\infty }}^0)$$$$\ln \mathrm{L}=\underset{x\to +\mathrm{\infty }}{\lim }\frac{1}{x}\ln x(\to 0\mathrm{\infty })$$$$\ln \mathrm{L}=\underset{x\to +\mathrm{\infty }}{\lim }\frac{\ln x}{x}(\to \frac{\mathrm{\infty }}{\mathrm{\infty }}){\mathrm{L}}^{\prime }\mathrm{hospital}$$$$\ln \mathrm{L}=\underset{x\to +\mathrm{\infty }}{\lim }\frac{1}{x}=0$$$$\mathrm{L}=1$$$$-----------$$$$\mathrm{D}=\underset{x\to 0^{+}}{\lim }{x}^{\frac{1}{x}}$$$$\ln \mathrm{D}=\underset{x\to 0^{+}}{\lim }\frac{1}{x}\ln x(\to +\mathrm{\infty }\times -\mathrm{\infty })$$$$\ln \mathrm{D}=-\mathrm{\infty }$$$$\mathrm{D}=0$$

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