Calculate-lim-x-0-1-x-1-x-3-x-

Question Number 168055 by qaz last updated on 01/Apr/22

Calculate :: \lim_{x \to 0} \frac{{(1 + x)}^{- \frac{1}{x^{3}}}}{x} = ?

Answered by LEKOUMA last updated on 03/Apr/22

\lim_{x \to 0} \frac{e^{((1 + x) - 1) \frac{1}{x^{3}}}}{x} = \lim_{x \to 0} \frac{e^{\frac{1}{x^{2}}}}{x} = + \infty

Answered by Mathspace last updated on 04/Apr/22

f (x) = \frac{{(1 + x)}^{- \frac{1}{x^{3}}}}{x} \Rightarrow

f (x) = \frac{1}{x} e^{- \frac{1}{x^{3}} l n (1 + x)}

{l n}^{'} (1 + x) = \frac{1}{1 + x} = 1 - x + o (x^{2})

\Rightarrow l n (1 + x) \sim x \Rightarrow

- \frac{1}{x^{3}} l n (1 + x) \sim - \frac{1}{x^{2}} (x \to 0) \Rightarrow

f (x) \sim \frac{1}{x} e^{- \frac{1}{x^{2}}} (\frac{1}{x} = y)

(x \to o^{+} \Rightarrow y \to + \infty) \Rightarrow

l i m f (x) = {l i m}_{y \to + \infty} {y e}^{- y^{2}} = 0