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

Question Number 147687 by mathmax by abdo last updated on 22/Jul/21

calculate \lim_{n \to 0} \frac{e^{- {nx}^{2}} - nx - 1}{x^{3}}

Answered by ArielVyny last updated on 23/Jul/21

e^{U_{n}} - 1 \sim U_{n} (U_{n} \to 0)

{l i m}_{n \to 0} \frac{e^{- {n x}^{2}} - 1 - n x}{x^{3}} = {l i m}_{n \to 0} \frac{- {n x}^{2} - n x}{x^{3}} = 0

\frac{e^{- {n x}^{2}} - n x - 1}{x^{3}} = \frac{\frac{1}{e^{{n x}^{2}}} - n x - 1}{x^{3}} = \frac{1}{x^{3} e^{{n x}^{2}}} - \frac{n x}{x^{3}} - \frac{1}{x^{3}} = \frac{1}{x^{3}} - \frac{1}{x^{3}} - \frac{n}{x^{2}} = 0 (n \to 0 {

Answered by mathmax by abdo last updated on 23/Jul/21

e^{- {nx}^{2}} \sim 1 - {nx}^{2} (n \to 0) \Rightarrow e^{- {nx}^{2}} - nx - 1 \sim 1 - {nx}^{2} - nx - 1

= - {nx}^{2} - nx \Rightarrow \frac{e^{- {nx}^{2}} - nx - 1}{x^{3}} \sim \frac{- {nx}^{2} - nx}{x^{3}} = - \frac{n}{x} - \frac{n}{x^{2}} \to 0 \Rightarrow

\lim_{n \to 0} \frac{e^{- {nx}^{2}} - nx - 1}{x^{2}} = 0