Using L'Hôpital's rule, evaluate lim of xe^(-x) as x approaches infinity

Answer 1

X=Lim xe^(-x)
x->∞
=Lim x/e^x
x->∞

L'hôpital's rule is applicable if direct evaluation results in an indeterminate form, which is the case here.
To apply the rule, differentiate both numerator and denominator to give
X=Lim 1/e^x
x->∞
=0