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

[Painguy]

Homework Statement

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)

Homework Equations

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 I am sure my answer is right.

 

[SammyS]

Homework Statement

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)

Homework Equations

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 I am sure my answer is right.

You have x = 1/u,

so if x → 0⁺, then so does 1/u → 0⁺.

What that implies is that u → +∞ .

 

[Painguy]

You have x = 1/u,

so if x → 0⁺, then so does 1/u → 0⁺.

What that implies is that u → +∞ .

That makes more sense. Thank you very much.