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.

Asked on by cspanutius

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lfryerda | High School Teacher | (Level 2) Educator

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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`

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