Question

What is the limit of `(1+x)^(1/x)` as x approaches 0?

Answer 1

`e`

Explanation:

From the binomial expansion

`(1+x)^n = 1 + nx + (n(n-1))/(2!)x^2+(n(n-1)(n-2))/(3!)x^3+ cdots +`

we have

`(1+x)^(1/x) = 1 + 1/x x + (1/x(1/x-1))/(2!)x^2+(1/x(1/x-1)(1/x-2))/(3!)x^3+ cdots +`

`=1+1+(1(1-x))/(2!)+(1(1-x)(1-2x))/(3!)+cdots+`

so

`lim_(x->0)(1+x)^(1/x) =lim_(x->0)1+1+(1(1-x))/(2!)+(1(1-x)(1-2x))/(3!)+cdots+=sum_(k=0)^oo1/(k!) = e`