Proof of a limit involving definition of differentiability

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Homework Help Overview

The problem involves proving a limit related to the definition of differentiability for a function at a specific point, specifically at x=0. The limit in question is as x approaches 0 for the expression involving the function evaluated at x squared minus the function evaluated at 0, divided by x.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants express confusion about manipulating the definition of differentiability and seek clarification on the limit. One participant suggests using L'Hôpital's Rule, while another notes that they cannot apply it due to their current coursework. A change of variable is proposed as an alternative approach.

Discussion Status

The discussion is ongoing, with participants exploring different methods to approach the problem. Some guidance has been offered regarding a change of variable, but there is no consensus on a specific method to proceed with the proof.

Contextual Notes

Participants mention constraints related to their coursework, specifically that L'Hôpital's Rule has not yet been proved, which limits their options for solving the problem.

mrchris
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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.
 
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mrchris said:

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.
 
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.
 
mrchris said:
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.
 

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