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Question Number 1767 by Rasheed Ahmad last updated on 18/Sep/15

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

Answered by 123456 last updated on 19/Sep/15

$$\mathrm{L}=\underset{x\to 0}{\lim }{\left(\frac{1}{x}\right)}^x$$$$\ln \mathrm{L}=\underset{x\to 0}{\lim }x\ln \left(\frac{1}{x}\right)=-\underset{x\to 0}{\lim }x\ln x$$$$\ln \mathrm{L}=-\underset{x\to 0}{\lim }\frac{\ln x}{\frac{1}{x}}(\to \frac{\mathrm{\infty }}{\mathrm{\infty }})$$$$\ln \mathrm{L}=-\underset{x\to 0}{\lim }\frac{\frac{1}{x}}{-\frac{1}{x^2}}=\underset{x\to 0}{\lim }x=0$$$$\mathrm{L}=1$$$$---------------$$$$\mathrm{D}=\underset{x\to +\mathrm{\infty }}{\lim }{\left(\frac{1}{x}\right)}^x$$$$\ln \mathrm{D}=-\underset{x\to +\mathrm{\infty }}{\lim }x\ln x(\to \mathrm{\infty }\times \mathrm{\infty })$$$$\ln \mathrm{D}=-\mathrm{\infty }$$$$\mathrm{D}=0$$

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