Question

What is the limit of `((e^x)-x)^(2/x)` as x approaches infinity?

Answer 1

`e^2`

Explanation:

`(e^x-x)^(2/x)=e^2(1-x/e^x)^(2/x)`

by the binomial expansion

`(1-x/e^x)^(2/x)=1+2/x(-x/e^x)+2/x(2/x-1)(-x/e^x)^2 1/(2!)+cdots`
`=1-2/e^x+(p_2(x))/e^(2x)+cdots+(-1)^n(p_n(x))/e^(nx)1/(n!)+cdots` where `p_n(x)` is a `n` degree polynomial.

We know that `lim_(x->oo)(p_n(x))/e^x = 0` so

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