Proving Simple Limit Statement: f(x) & f(a+h)

[mscbuck]

Homework Statement

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)"

Homework Equations

The Attempt at a Solution

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

Thanks!

 

[JonF]

x + a - a = x correct? Let's h = x - a. that means:

x = h + a
h = x - a

also you should know this fact, lim x-->a is the smae as x+b-->a+b
For example the limit as x tends to 2, is the same as the limit as x+2 tends to 4.
so x->a is the same as x-a->a-a.

Sub these into: f(x) as x-->a
and you get f(h+a) as x-a -> a-a
which is equivlent to f(h+a) as h-> 0

 

[fzero]

You're probably expected to go back to the \epsilon,\delta definition of the limit and use those techniques.

 

[JonF]

figured he didn't have to do that since he said

this is really just an exercise in understanding what the terms are

but a change of variables is rigorous and I'm pretty sure if you went back to δ ε you'd just do the same idea there.