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Question Number 70364 by 20190927 last updated on 03/Oct/19

$$\mathrm{solve}\mathrm{L}=\underset{x\to 0}{\lim }\frac{{\mathrm{e}}^{\mathrm{x}}-\frac{1}{1-\mathrm{x}}}{{\mathrm{x}}^2}$$

Commented by kaivan.ahmadi last updated on 03/Oct/19

$${lim}_{x\to 0}\frac{(1-x)e^x-1}{x^2(1-x)}={lim}_{x\to 0}\frac{-{xe}^x}{2x-3x^2}=$$$${lim}_{x\to 0}\frac{-e^x-{xe}^x}{2-6x}=\frac{-1}{2}$$

Commented by 20190927 last updated on 03/Oct/19

$$\mathrm{Thank}\mathrm{you}$$

Commented by mathmax by abdo last updated on 03/Oct/19

$$letf\left(x\right)=\frac{e^x-\frac{1}{1-x}}{x^2}\Rightarrow f\left(x\right)=\frac{(1-x)e^x-1}{x^2(1-x)}$$$$wehave{e}^x=1+x+\frac{x^2}{2}+o\left(x^3\right)\Rightarrow (1-x)e^x=(1-x)(1+x+\frac{x^2}{2}+o\left(x^3\right))$$$$=1+x+\frac{x^2}{2}-x-x^2-\frac{x^3}{2}+o\left(x^4\right)=1-\frac{x^2}{2}-\frac{x^3}{2}+o\left(x^4\right)\Rightarrow$$$$f\left(x\right)\sim \frac{-\frac{x^2}{2}-\frac{x^3}{2}}{x^2(1-x)}\Rightarrow f\left(x\right)\sim \frac{-\frac{1}{2}-\frac{x}{2}}{1-x}(x\to 0)\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=-\frac{1}{2}$$

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