# Understanding the proof of Excercise 10b from chapter 5 of Spivak’s Calculus.

The question is “Prove that the limit as x approaches zero of f(x)=the limit as x approaches a of f(x-a).

I have taken a few stabs at the problem and was told that the solution was the image I attached.

The given proof largely makes sense to me, only it seems to be a proof that the limit as x approaches a of f(x) equals the limit as y approaches zero of g(y-a). I don’t see how this is a proof of what was asked, and I also don’t see how to proceed in the other direction. I think my issue is that I’m struggling to connect my intuitive/verbal understanding with epsilon-delta limit notation, and changing the variable from x to y also confuses me. Can anyone help make sense of this?

$$\lim_{x\to 0}f(x)=L\iff$$ $$\forall e>0\,\exists d>0\, \forall x\,(0<|x| $$\forall e>0\, \exists d>0\,\forall x'\,(0<|x'-a| $$\lim_{x'\to a}f(x'-a)=L \iff$$ $$\lim_{x\to a}f(x-a)=L.$$ Regardless of whether or not $$a$$ is a $$0$$ of $$f.$$
• You can also look at MathJax.Org. This site has MathJax in it. E.g. typing a dollar sign before and after \lim_{x\to 0}f(x)=L\iff gives $\lim_{x\to 0}f(x)=L\iff$. – DanielWainfleet Oct 20 '19 at 7:33