# 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.

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### 1 Answer

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`