# Find limits for lim x-->∞  xe^-x  Use L'Hospital's Rule when appropriate. If there is a more elementary algebraic method, you might consider it. If you are able to use L' Hopital's Rule and...

Find limits for lim x-->∞  xe^-x  Use L'Hospital's Rule when appropriate. If there is a more elementary algebraic method, you might consider it.

If you are able to use L' Hopital's Rule and when you convert your expression to a fraction, check and state that the hypotheses of the theorem are satifsied. check problem by graphing the function near the limit and see if the values support your solution.

lfryerda | Certified Educator

Direct substitution of this limit shows that it is in \infty\cdot 0 form.  To use L'Hopital's rule, the limit needs to be in 0/0 form.  First the limit needs to be rearranged into the form for L'Hopital's rule, then you can apply the rule by taking derivatives of numerator and denominator.

lim_{x->\infty}xe^{-x}

=lim_{x->\infty}x/e^x    rearranging to be able to use L'Hopital's rule

=lim_{x->\infty}1/e^x    apply L'Hopital's rule with taking derivatives

=1/\infty

=0