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1. The problem statement, all variables and given/known data

1)Prove that > limit(f(x), x = a) = limit(f(a+h), h = 0)

2. Prove that limit(f(x), x = a) = L iff limit(f(x)-L, x = a) = 0

2. Relevant equations

3. The attempt at a solution

1)I have tried to use the definition |f(x)-limf(a+h)| and |f(a+h)-limf(x)|. But it doesnt seem to be working because no further simplification can be done. I cant find a way to relate it back to |x-2|<delta and |h|<delta because the i cant eliminate the limit of function. I have also tried to assume both limit equals to f(a) and proves them. But again, it stucks.

2)First I assume that limit(f(x), x = a) = L. Then, the definition follows.

There exists 0<|x-a|<delta such that |f(x)-L|< epsilon.....(1)

So, in the case of limit(f(x)-L, x = a) = 0.

0<|x-a|<delta |f(x)-L-0|=|f(x)-L|<epsilon.... from (1)

and vice versa.

But it doesnt look like a proof to me. It more like I am rewriting it in another way. Is there a better way to put it?

Thanks...

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# Homework Help: Definition of Limit

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