Question

What is the limit of `(x^n)*(e^-x)` as x approaches infinity?

Answer 1

`lim_{x->infty}x^n e^{-x}= 0`

Explanation:

We know that `e^x = sum_{nu=0}^{infty}(x^nu)/(nu!)` so
`x^n e^{-x} = (x^n)/(e^{x})=(x^n)/(sum_{i=0}^{i=n}(x^i)/(i!)+sum_{i=n+1}^{infty}(x^i)/(i!)`. We conclude that
`lim_{x->infty}(x^n)/(e^{x}) = 0`