# Use L'Hopital's Rule to relate to limit definition for e

1. Nov 6, 2012

### Painguy

1. The problem statement, all variables and given/known data
It can be shown that
lim
n→∞(1 + 1/n)^n = e.
Use this limit to evaluate the limit below.

lim
x→0+ (1 + x)^(1/x)

2. Relevant equations

3. The attempt at a solution
So i guess what i need to do is try to get that limit in the form of the limit definition for e.

lim
x→0+ (1 + x)^(1/x)

x=1/u

since x-> 0 that means 1/u ->inf

lim
x→0+ (1 + 1/u)^(1/(1/u))

= lim
1/u→∞ (1 + 1/u)^(u) = e

I feel like my last 2 steps are wrong, but im sure my answer is right.

2. Nov 6, 2012

### SammyS

Staff Emeritus
You have x = 1/u,

so if x → 0+, then so does 1/u → 0+.

What that implies is that u → +∞ .

3. Nov 6, 2012

### Painguy

That makes more sense. Thank you very much.