Question

How do you find the limit of `e^x/x^3` as x approaches infinity?

Answer 1

Because it is in the indeterminate form `oo/oo` we can apply L'Hôpital's rule three times respectively to get

`lim (e^x/x^3)=lim (((e^x)')/((x^3)'))=lim (e^x)/(3x^2)=(oo/oo)`

`lim (((e^x)')/((3x^2)')) = lim (e^x)/(6x)= (oo/oo)`

`lim ((e^x)')/((6x)')= lim e^x/6=oo`

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