Question Number 75929 by Rio Michael last updated on 21/Dec/19
![Evaluate lim_(x→−∞) [(√(1−xe^x )) ]](Q75929.png)
$${Evaluate} \\ $$$$\underset{{x}\rightarrow−\infty} {\:\mathrm{lim}}\:\left[\sqrt{\mathrm{1}−{xe}^{{x}} }\:\right] \\ $$
Commented by kaivan.ahmadi last updated on 21/Dec/19
![lim_(x→−∞) xe^x =lim_(x→−∞) (x/e^(−x) )=lim_(x→−∞) (1/(−e^(−x) ))=0^− ⇒lim_(x→−∞) (√(1−xe^x ))=(√(1−0^− ))=(√1^+ )=1^+ ⇒lim_(x→−∞) [(√(1−xe^x ))]=[1^+ ]=1](Q75975.png)
$${lim}_{{x}\rightarrow−\infty} {xe}^{{x}} ={lim}_{{x}\rightarrow−\infty} \frac{{x}}{{e}^{−{x}} }={lim}_{{x}\rightarrow−\infty} \frac{\mathrm{1}}{−{e}^{−{x}} }=\mathrm{0}^{−} \\ $$$$\Rightarrow{lim}_{{x}\rightarrow−\infty} \sqrt{\mathrm{1}−{xe}^{{x}} }=\sqrt{\mathrm{1}−\mathrm{0}^{−} }=\sqrt{\mathrm{1}^{+} }=\mathrm{1}^{+} \\ $$$$\Rightarrow{lim}_{{x}\rightarrow−\infty} \left[\sqrt{\mathrm{1}−{xe}^{{x}} }\right]=\left[\mathrm{1}^{+} \right]=\mathrm{1} \\ $$
Commented by Rio Michael last updated on 21/Dec/19
$${thanks}\:{sir} \\ $$