# Simple Limit Proof/Reasoning

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• Sep 28th 2010, 03:50 PM
mscbuck
Simple Limit Proof/Reasoning
Hey all, thanks again for all help in the past! Here's a limit question that I am unsure if I am overthinking, or if it's as simple as it looks.

It states:

Prove:
lim f(x) as x-->a is equal to lim f(a+h) as h-->0 (this is really just an exercise in understanding what the terms are)

Now the easiest way would just to pop in the values and boom, lim f(a) = lim f(a). But I think he is looking for a different reasoning. My attempt was to kind of define the statements like so:

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LHS: f can be made to be as close to a limit L as desired by making x sufficiently close to a.

RHS: f can be made to be as close to a limit L as desired by making h sufficiently close to zero.

And in this example, it just so happens when h is made to go to zero we are left with the respective equations L's equal to each other.
______

I don't think I've really "proved" anything though, have I?

Thanks again!
• Sep 28th 2010, 08:51 PM
Prove It
It depends on whether $\displaystyle f(x)$ and $\displaystyle f(a + h)$ are continuous at $\displaystyle x = a$ and $\displaystyle h = 0$ respectively, and if you are allowed to use the result that $\displaystyle \lim_{x \to n}f(x) = f(n)$ if $\displaystyle f(x)$ is continuous at $\displaystyle x = n$.

If so, your reasoning is fine.

If not, you will need to use an $\displaystyle \epsilon -\delta$ proof.