Question

How do you find the limit of `(x ^ 3)(e ^ (-x ^ 2))` as x approaches infinity?

Answer 1

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

Explanation:

Write the limit as:

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

It is now in the indefinite form `oo/oo` and we can apply l'Hospital's rule:

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

and again:

`lim_(x->oo) (3x)/(2e^(x^2)) = lim_(x->oo) 3/(4xe^(x^2)) = 0`