`lim_(x->oo) x^3 e^(-x^2)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply,...

`lim_(x->oo) x^3 e^(-x^2)` Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.

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Chapter 4, 4.4 - Problem 45 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`lim_(x->oo)x^3e^(-x^2)`

`=lim_(x->oo)x^3/e^(x^2)`

Apply L'Hospital rule,

`=lim_(x->oo)((x^3)')/((e^(x^2))')`

`=lim_(x->oo)(3x^2)/(2xe^(x^2))`

`=lim_(x->oo)(3x)/(2e^(x^2))`

Again apply L'Hospital rule,

`=lim_(x->oo)((3x)')/((2e^(x^2))')`

`=lim_(x->oo)3/(2*2xe^(x^2))`

`=lim_(x->oo)(3*1/e^(x^2))/(4x)`

Now `lim_(x->oo)e^(x^2)=oo`

`lim_(x->oo)3/e^(x^2)=3/oo=0`

`lim_(x->oo)4x=oo`

`=0/oo`

`=0`

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