Lim of x^3 × e^(-x^2) as x approaches infinity?

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Answer 1 · 2018-06-05T08:22:37

#L=lim_(x->oo)x^3*e^(-x^2)=0#

We want to solve

#L=lim_(x->oo)x^3*e^(-x^2)=lim_(x->oo)x^3/e^(x^2)#

Which is an indeterminate form #oo/oo#

So we can apply L'Hôpital's rule

#color(blue)(lim_(x->c)f(x)/g(x)=lim_(x->c)(f'(x))/(g'(x))#

Thus

#L=lim_(x->oo)(3x^2)/(2xe^(x^2))=lim_(x->oo)(3x)/(2e^(x^2))#

Again an indeterminate form #oo/oo#, so apply LHR again

#L=lim_(x->oo)(3)/(4xe^(x^2))=0#