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?
$$\lim_{x\to 0}f(x)=L\iff$$ $$ \forall e>0\,\exists d>0\, \forall x\,(0<|x|<d\implies |f(x)-L|<e)\iff$$ $$ \forall e>0\, \exists d>0\,\forall x'\,(0<|x'-a|<d\implies |f((x'-a))-L|<e) \iff$$ $$ \lim_{x'\to a}f(x'-a)=L \iff$$ $$ \lim_{x\to a}f(x-a)=L.$$ Regardless of whether or not $a$ is a $0$ of $f.$