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Find-Lim-x-e-x-x-




Question Number 148326 by Odhiambojr last updated on 27/Jul/21
Find   Lim_(x→∞) (e^x +x)
$${Find}\: \\ $$$${Li}\underset{{x}\rightarrow\infty} {{m}}\left({e}^{{x}} +{x}\right) \\ $$
Answered by mathmax by abdo last updated on 27/Jul/21
lim_(x→+∞) x+e^x  =lim_(x→+∞) e^x (xe^(−x)  +1)=lim_(x→+∞) e^x  =+∞  because lim_(x→+∞) xe^(−x) =0  lim_(x→−∞) x+e^x  =lim_(x→−∞) x +lim_(x→−∞) e^(−x)   =−∞+0=−∞
$$\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{x}+\mathrm{e}^{\mathrm{x}} \:=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{e}^{\mathrm{x}} \left(\mathrm{xe}^{−\mathrm{x}} \:+\mathrm{1}\right)=\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{e}^{\mathrm{x}} \:=+\infty \\ $$$$\mathrm{because}\:\mathrm{lim}_{\mathrm{x}\rightarrow+\infty} \mathrm{xe}^{−\mathrm{x}} =\mathrm{0} \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow−\infty} \mathrm{x}+\mathrm{e}^{\mathrm{x}} \:=\mathrm{lim}_{\mathrm{x}\rightarrow−\infty} \mathrm{x}\:+\mathrm{lim}_{\mathrm{x}\rightarrow−\infty} \mathrm{e}^{−\mathrm{x}} \\ $$$$=−\infty+\mathrm{0}=−\infty \\ $$

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