Proof of a limit involving definition of differentiability

[mrchris]

Homework Statement

let the function f:ℝ→ℝ be differentiable at x=0. Prove that lim x→0
[f(x²)-f(0)]
______________ =0
x

Homework Equations

The Attempt at a Solution

I am kind of lost on this one, I have tried manipulating the definition of a differentiable function at x=0 and I am not making much progress. I am not just looking for an answer here, this actually a review problem for a test and I am really trying to understand it so any help would be appreciated.

 

[Curious3141]

Homework Statement

let the function f:ℝ→ℝ be differentiable at x=0. Prove that lim x→0
[f(x²)-f(0)]
______________ =0
x

Homework Equations

The Attempt at a Solution

I am kind of lost on this one, I have tried manipulating the definition of a differentiable function at x=0 and I am not making much progress. I am not just looking for an answer here, this actually a review problem for a test and I am really trying to understand it so any help would be appreciated.

Try L' Hopital's Rule.

 

[mrchris]

I am in an analysis class and we have not yet proved L' Hopital's Rule so I can not use it in this proof.

 

[Dick]

I am in an analysis class and we have not yet proved L' Hopital's Rule so I can not use it in this proof.

Use the change of variable u=x^2. Now think about the limit as u->0.