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




Question Number 90226 by manuel__ last updated on 22/Apr/20
lim_(x→0) (((x^3 (e^x −1)−x^2 )/(xe^x −x)))=?
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{x}^{\mathrm{3}} \left({e}^{{x}} −\mathrm{1}\right)−{x}^{\mathrm{2}} }{{xe}^{{x}} −{x}}\right)=? \\ $$
Commented by john santu last updated on 22/Apr/20
lim_(x→0)  ((x^2 (e^x −1)−x)/(e^x −1)) =   lim_(x→0)  ((2x(e^x −1)+x^2 e^x −1)/e^x ) = −1
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} \left({e}^{{x}} −\mathrm{1}\right)−{x}}{{e}^{{x}} −\mathrm{1}}\:=\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{x}\left({e}^{{x}} −\mathrm{1}\right)+{x}^{\mathrm{2}} {e}^{{x}} −\mathrm{1}}{{e}^{{x}} }\:=\:−\mathrm{1} \\ $$
Commented by mathmax by abdo last updated on 22/Apr/20
let f(x)=((x^3 (e^x −1)−x^2 )/(xe^x −x)) ⇒f(x)=((x^2 (e^x −1)−x)/(e^x −1))  e^x  ∼1+x ⇒x^2 (e^x −1)−x ∼x^3 −x ⇒f(x) ∼((x^3 −x)/x) =x^2 −1 ⇒  lim_(x→0)   f(x)=−1
$${let}\:{f}\left({x}\right)=\frac{{x}^{\mathrm{3}} \left({e}^{{x}} −\mathrm{1}\right)−{x}^{\mathrm{2}} }{{xe}^{{x}} −{x}}\:\Rightarrow{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} \left({e}^{{x}} −\mathrm{1}\right)−{x}}{{e}^{{x}} −\mathrm{1}} \\ $$$${e}^{{x}} \:\sim\mathrm{1}+{x}\:\Rightarrow{x}^{\mathrm{2}} \left({e}^{{x}} −\mathrm{1}\right)−{x}\:\sim{x}^{\mathrm{3}} −{x}\:\Rightarrow{f}\left({x}\right)\:\sim\frac{{x}^{\mathrm{3}} −{x}}{{x}}\:={x}^{\mathrm{2}} −\mathrm{1}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\:{f}\left({x}\right)=−\mathrm{1} \\ $$

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