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Question

$\underset{x	o 0}{lim}\frac{{e}^{2|\sin x|}-2|\sin x|-1}{{x}^{2}}$

Accepted Answer

$\begin{gathered} \underset{x	o 0}{lim}\frac{{e}^{2|\sin x|}-2|\sin x|-1}{{x}^{2}} \ \underset{x	o 0}{lim}\frac{{e}^{2|\sin x|}-2|\sin x|-1}{|\sin x{|}^{2}}	imes \frac{{\sin }^{2}x}{{x}^{2}} \end{gathered}$

Let $|\sin x|=t$

$\begin{gathered} \underset{t	o 0}{lim}\frac{{e}^{2t}-2t-1}{{t}^{2}}	imes \underset{x	o 0}{lim}\frac{{\sin }^{2}x}{{x}^{2}} \ =\underset{t	o 0}{lim}\frac{2{e}^{2t}-2}{2t}	imes 1=2	imes 1=2 \end{gathered}$

Solutions · 1