The discussion revolves around proving a limit property for even functions, specifically that for a function f: R->R, the limit as x approaches 0 of f(x) equals L if and only if the limit as x approaches 0 from the positive side of f(x) equals L.
Participants are actively engaging with the definitions of limits and attempting to clarify the connections between the limits from the positive and negative sides. Some have expressed confusion regarding the organization of their arguments and the application of definitions.
There is a focus on the definitions of limits as they relate to the even function property, with participants questioning how to effectively utilize these definitions in their proof. The variable 'a' is noted to be 0 in the context of the limits being discussed.
MathSquareRoo said:Homework Statement
Prove that if f: R->R is an even function, then lim x->0 f(x)=L if and only if lim x->0+ f(x)=L.
Homework Equations
The Attempt at a Solution
So far I have:
If f is an even function f(x)=f(-x) for x in domain of f.
Then I am trying to apply the limit definitions, but am unsure of how to write the proof from here.
MathSquareRoo said:So lim x->0+ f(x)=L implies there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(x)-Ll<epsilon provided 0<x-a<delta.
Then lim x->0+ f(-x)=L implies that there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(-x)-Ll<epsilon provided 0<x-a<delta.
I have the definitions, but I don't understand the last part of your comment, can you clarify?