Determine-i-lim-a-1-a-a-ii-lim-a-0-1-a-a-

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

D e t e r m i n e

(i) \underset{a \to \infty}{l i m} {(\frac{1}{a})}^{a} (i i) \underset{a \to 0}{l i m} {(\frac{1}{a})}^{a}

Answered by 123456 last updated on 19/Sep/15

L = \lim_{x \to 0} {(\frac{1}{x})}^{x}

\ln L = \lim_{x \to 0} x \ln (\frac{1}{x}) = - \lim_{x \to 0} x \ln x

\ln L = - \lim_{x \to 0} \frac{\ln x}{\frac{1}{x}} (\to \frac{\infty}{\infty})

\ln L = - \lim_{x \to 0} \frac{\frac{1}{x}}{- \frac{1}{x^{2}}} = \lim_{x \to 0} x = 0

L = 1

- - - - - - - - - - - - - - -

D = \lim_{x \to + \infty} {(\frac{1}{x})}^{x}

\ln D = - \lim_{x \to + \infty} x \ln x (\to \infty \times \infty)

\ln D = - \infty

D = 0

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