# [Spivak Calculus, Ch. 5 P. 9] Showing equality of two limits

1. Mar 30, 2013

### WisheDeom

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

Prove that $\lim_{x \rightarrow a} f(x) = \lim_{h \rightarrow 0} f(a+h)$.

2. Relevant equations

By definition, if $\lim_{x \rightarrow a} f(x) = l$ then for every $\epsilon > 0$ there exists some $\delta_1$ such that for all x, if $0<|x-a|<\delta_1$ then $|f(x)-l|<\epsilon$.

Similarly, if $\lim_{h \rightarrow 0} f(a+h) = m$ then for every $\epsilon > 0$ there exists some $\delta_2$ such that for all h, if $0<|h-0|<\delta_2$ then $|f(a+h)-m|<\epsilon$.

3. The attempt at a solution

I'm really not sure how to go from here. I think maybe I need to perform a proof by contradiction by assuming that $l \neq m$, but I don't know what kind of contradiction I'm looking for.

Last edited: Mar 30, 2013
2. Mar 30, 2013

### voko

I think you need to prove that if the LHS has some limit l, then the RHS also has that same limit, and vice versa.

3. Mar 30, 2013

### E'lir Kramer

Edit: actually, not quite sure what he's asking for here.