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

Question Number 76191 by abdomathmax last updated on 25/Dec/19

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

Answered by Rio Michael last updated on 25/Dec/19

A m n o t s u r e i f t h i s l i m i t

w o u l d e x i s t b e c a u s e

\underset{x \to 0^{-}}{l i m} \frac{e^{x} - e^{[x]}}{x} = 1 a s e^{[x]} \to 0 a s x \to 0

\underset{x \to 0^{+}}{l i m} \frac{e^{x} - e^{[x]}}{x} = 0

Commented by MJS last updated on 25/Dec/19

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

- 1 ⩽ x < 0 \Rightarrow e^{[x]} = \frac{1}{e}

\Rightarrow as x \to 0^{-} we get \frac{1 - \frac{1}{e}}{x \to 0^{-}} = - \infty

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