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Question Number 101096 by student work last updated on 30/Jun/20

find the oblique asymptote of f(x)=x∙e^(1/x) i need your help

$$\mathrm{find}\:\mathrm{the}\:\mathrm{oblique}\:\mathrm{asymptote}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\centerdot\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} \:\:\:\:\: \\ $$$$\:\mathrm{i}\:\mathrm{need}\:\mathrm{your}\:\mathrm{help} \\ $$

Commented by student work last updated on 30/Jun/20

i need soon

$$\mathrm{i}\:\mathrm{need}\:\mathrm{soon} \\ $$

Commented by student work last updated on 30/Jun/20

who can help me?

$$\mathrm{who}\:\mathrm{can}\:\mathrm{help}\:\mathrm{me}? \\ $$

Answered by mathmax by abdo last updated on 30/Jun/20

lim_(x→+∞) f(x) =lim_(x→+∞) xe^(1/x) =lim_(x→+∞) x =+∞ lim_(x→+∞) ((f(x))/x) =lim_(x→+∞) e^(1/x) =1 lim_(x→+∞) f(x)−x =lim_(x→+∞) x e^(1/x) −x =lim_(x→+∞) x(e^(1/x) −1) but e^(1/x) =1+(1/x) +o((1/x^2 )) ⇒e^(1/x) −1 =(1/x)+o((1/x^2 )) ⇒x(e^(1/x) −1) =1+o((1/x)) ⇒ lim_(x→+∞) f(x)−x =1 ⇒the line y =x+1 is oblique assymptote to C_f at +∞ same result at −∞

$$\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \:\mathrm{xe}^{\frac{\mathrm{1}}{\mathrm{x}}} \:=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{x}\:=+\infty \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \:\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x}}\:=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \:\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} \:=\mathrm{1} \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{f}\left(\mathrm{x}\right)−\mathrm{x}\:=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{x}\:\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} −\mathrm{x}\:=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{x}\left(\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} −\mathrm{1}\right)\:\mathrm{but} \\ $$$$\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} \:=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}}\:+\mathrm{o}\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\:\Rightarrow\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} −\mathrm{1}\:=\frac{\mathrm{1}}{\mathrm{x}}+\mathrm{o}\left(\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\right)\:\Rightarrow\mathrm{x}\left(\mathrm{e}^{\frac{\mathrm{1}}{\mathrm{x}}} −\mathrm{1}\right)\:=\mathrm{1}+\mathrm{o}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:\Rightarrow \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{f}\left(\mathrm{x}\right)−\mathrm{x}\:=\mathrm{1}\:\Rightarrow\mathrm{the}\:\mathrm{line}\:\mathrm{y}\:=\mathrm{x}+\mathrm{1}\:\mathrm{is}\:\mathrm{oblique}\:\mathrm{assymptote}\:\:\mathrm{to}\:\mathrm{C}_{\mathrm{f}} \:\mathrm{at}\:+\infty \\ $$$$\mathrm{same}\:\mathrm{result}\:\mathrm{at}\:−\infty \\ $$$$ \\ $$

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