lim-x-0-1-x-1-x-e-1-x-

Question Number 96319 by M±th+et+s last updated on 31/May/20

\underset{x \to 0}{l i m} {(\frac{{(1 + x)}^{\frac{1}{x}}}{e})}^{\frac{1}{x}}

Answered by abdomathmax last updated on 31/May/20

f (x) = {(\frac{{(1 + x)}^{\frac{1}{x}}}{e})}^{\frac{1}{x}} \Rightarrow \ln (f (x)) = \frac{1}{x} \ln (\frac{{(1 + x)}^{\frac{1}{x}}}{e})

= \frac{1}{x} (\ln {(1 + x)}^{\frac{1}{x}} - 1)

= \frac{1}{x} (\frac{1}{x} \ln (1 + x) - 1) = \frac{\ln (1 + x) - x}{x^{2}}

hodpiral \to \lim_{x \to 0} \frac{\ln (1 + x) - x}{x^{2}}

= \lim_{x \to 0} \frac{\frac{1}{1 + x} - 1}{2 x} = \lim_{x \to 0} \frac{- 1}{2 (1 + x^{2})} = - \frac{1}{2} \Rightarrow

\ln (f (x)) \to - \frac{1}{2} \Rightarrow f (x) \to e^{- \frac{1}{2}} = \frac{1}{\sqrt{e}}

Commented by M±th+et+s last updated on 31/May/20

n i c e w o r k s i r t h a n k y o u

Commented by mathmax by abdo last updated on 31/May/20

you are welcome sir .