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.
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.
The definition of differentiability is a mathematical concept that describes the smoothness of a function. A function is said to be differentiable at a point if the slope of the tangent line at that point exists and is unique.
Differentiability implies continuity, but continuity does not necessarily imply differentiability. A function can be continuous at a point without being differentiable at that point. However, a function must be continuous to be differentiable.
The limit is involved in the definition of differentiability because it is used to determine the slope of the tangent line, which is necessary for a function to be differentiable at a point. The limit is used to find the derivative of a function, which is the slope of the tangent line.
A one-sided limit only considers the values of a function approaching a specific point from one direction, either the left or the right. A two-sided limit considers the values of a function approaching a specific point from both directions.
No, a function cannot be differentiable at a point if it is not continuous at that point. Differentiability requires continuity, so if a function is not continuous at a point, it cannot be differentiable at that point.