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Question Number 1767 by Rasheed Ahmad last updated on 18/Sep/15
Determine   (i) lim_(a→∞)  ((1/a))^a    (ii)  lim_(a→0)  ((1/a))^a
$${Determine} \\ $$$$\:\left({i}\right)\:\underset{{a}\rightarrow\infty} {{lim}}\:\left(\frac{\mathrm{1}}{{a}}\right)^{{a}} \:\:\:\left({ii}\right)\:\:\underset{{a}\rightarrow\mathrm{0}} {{lim}}\:\left(\frac{\mathrm{1}}{{a}}\right)^{{a}} \: \\ $$
Answered by 123456 last updated on 19/Sep/15
L=lim_(x→0) ((1/x))^x   ln L=lim_(x→0)  xln((1/x))=−lim_(x→0)  xln x  ln L=−lim_(x→0) ((ln x)/(1/x))        (→(∞/∞))  ln L=−lim_(x→0) ((1/x)/(−(1/x^2 )))=lim_(x→0)  x=0  L=1  −−−−−−−−−−−−−−−  D=lim_(x→+∞) ((1/x))^x   ln D=−lim_(x→+∞)  xln x     (→∞×∞)  ln D=−∞  D=0
$$\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}}\right)^{{x}} \\ $$$$\mathrm{ln}\:\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}\mathrm{ln}\left(\frac{\mathrm{1}}{{x}}\right)=−\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}\mathrm{ln}\:{x} \\ $$$$\mathrm{ln}\:\mathrm{L}=−\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{ln}\:{x}}{\frac{\mathrm{1}}{{x}}}\:\:\:\:\:\:\:\:\left(\rightarrow\frac{\infty}{\infty}\right) \\ $$$$\mathrm{ln}\:\mathrm{L}=−\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{{x}}}{−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}=\mathrm{0} \\ $$$$\mathrm{L}=\mathrm{1} \\ $$$$−−−−−−−−−−−−−−− \\ $$$$\mathrm{D}=\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}}\right)^{{x}} \\ $$$$\mathrm{ln}\:\mathrm{D}=−\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{x}\mathrm{ln}\:{x}\:\:\:\:\:\left(\rightarrow\infty×\infty\right) \\ $$$$\mathrm{ln}\:\mathrm{D}=−\infty \\ $$$$\mathrm{D}=\mathrm{0} \\ $$

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