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## Homework Statement

Prove that lim

_{x[itex]\rightarrow[/itex]c}f(x)=L if and only if lim

_{h[itex]\rightarrow[/itex]0}f(x+h)=L.

## Homework Equations

## The Attempt at a Solution

I think this is a simple problem, but I am getting caught up in the middle, as I'm not sure if my procedure is a valid way to prove the statement.

Suppose lim

_{x[itex]\rightarrow[/itex]c}f(x)=L.

Then for each [itex]\epsilon[/itex]>0 there is a [itex]\delta[/itex]>0 so that

if |x-c|<[itex]\delta[/itex] then |f(x)-L|<[itex]\epsilon[/itex].

Let x=c+h. Then if |x-c|<δ [itex]\Rightarrow[/itex] |c+h-c|<δ [itex]\Rightarrow[/itex] |h|<δ. Furthermore, |f(c+h)-L|<ε. And since we assumed lim

_{x[itex]\rightarrow[/itex]c}f(x)=L, it follows then that lim

_{h[itex]\rightarrow[/itex]0}f(c+h)=L.*

So my questions is this: it is valid to simply let x=c+h as I did?

*Note, I did not prove the other case, but I just wrote this out so you guys can get a good idea of what my argument is and tell me if it is wrong or not.