find-lim-x-0-e-x-x-x-

Question Number 38111 by maxmathsup by imad last updated on 21/Jun/18

f i n d {l i m}_{x \to 0} \frac{e^{x} - [x]}{x}

Commented by math khazana by abdo last updated on 22/Jun/18

t h e Q i s f i n d {l i m}_{x \to 0} \frac{e^{x} - 1 - [x]}{x}

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jun/18

[x] = 0 w h e n x \to 0 +

\lim_{x \to 0 +} \frac{e^{x} - [x]}{x}

= \lim_{x \to 0 +} \frac{e^{x}}{x} = \infty

[x] \to - 1 w h e n x \to 0 -

\lim_{x \to 0 -} \frac{e^{x} - [x]}{x}

= \lim_{x \to 0 -} \frac{e^{x} - 1}{x} = 1

i f

\lim_{x \to 0} \frac{e^{x} - 1 - [x]}{x}

\lim_{x \to 0 -} \frac{e^{x} - 1 + 1}{x} {[x] \to - 1 w h e n x \to 0 -

= \lim_{x \to 0 -} \frac{e^{x}}{x} = \infty

\underset{x \to 0 -}{li}

\lim_{x \to 0 +} \frac{e^{x} - 1 - [x]}{x}

= \lim_{x \to 0 +} \frac{e^{x} - 1 - 0}{x}

= 1