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lim-x-0-x-2-e-x-1-x-




Question Number 212097 by liuxinnan last updated on 30/Sep/24
lim_(x→0) (x^2 +e^x )^(1/x) =?
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left({x}^{\mathrm{2}} +{e}^{{x}} \right)^{\frac{\mathrm{1}}{{x}}} =? \\ $$
Answered by mehdee7396 last updated on 30/Sep/24
lim_(x→0)  ((x^2 +e^x −1)/x)=^(hop) lim_(x→0) ((2x+e^x )/1)=1  ⇒ans=e^1 =e ✓
$${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{x}^{\mathrm{2}} +{e}^{{x}} −\mathrm{1}}{{x}}\overset{{hop}} {=}{lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{2}{x}+{e}^{{x}} }{\mathrm{1}}=\mathrm{1} \\ $$$$\Rightarrow{ans}={e}^{\mathrm{1}} ={e}\:\checkmark \\ $$$$ \\ $$

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