calculate-lim-x-0-e-3x-2-1-x-2-

Question Number 33312 by abdo imad last updated on 14/Apr/18

c a l c u l a t e {l i m}_{x \to 0} \frac{e^{- 3 x^{2}} - 1}{x^{2}} .

Answered by Joel578 last updated on 14/Apr/18

\lim_{x \to 0} \frac{e^{- 3 x^{2}} - 1}{x^{2}} (L^{'} H o p i t a l)

= \lim_{x \to 0} \frac{- 3}{e^{3 x^{2}}} = - 3

Commented by prof Abdo imad last updated on 15/Apr/18

w e h a v e e^{u} = 1 + u + o (u^{2}) (u \to 0) \Rightarrow

e^{- 3 x^{2}} = 1 - 3 x^{2} + o (x^{4}) \Rightarrow e^{- 3 x^{2}} - 1 = - 3 x^{2} + o (x^{4})

\Rightarrow \frac{e^{- 3 x^{2}} - 1}{x^{2}} = - 3 + o (x^{2}) \Rightarrow {l i m}_{x \to 0} \frac{e^{- 3 x^{2}} - 1}{x^{2}} = - 3 .