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I have been trying to understand how this proof works to show $\lim_{x \to 0^+} f(x)$ = $\lim_{x \to +0^-} f(-x)$ from here; https://math.stackexchange.com/a/3399848/734140 $$ 0 < x < \delta \implies |f(x) - L| < \epsilon \implies -\delta < x < 0 \implies |f(x) - L|, $$ I don't understand how one can just switch the variables, i'm assuming he just negated the inequality and changed the variable of -x to x, but I was thinking that since -(-x) = x that means $0 < x < \delta \implies 0 < 0-(-x) < \delta$, then by definition of right hand limit, $\lim_{x \to 0^-} f(-x) = L$. Is this reasoning correct or how is the other post correct?

USER149372
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It's kinda like a change of variables, now, we want to prove the limit from below supposing that we have the limit from above, notice that the definition of limit from below of $f(-x)$ is that for every $\epsilon>0 $ there exists $\delta'>0$ so that $$ 0<-x<\delta'\implies |f(-x) -L| < \epsilon $$ but the definition of the limit from above is $$0 <x < \delta \implies |f(x)-L| < \epsilon$$ by the definitions provided by the OP. Now suppose that we have the latter, take $y = -x$ now suppose $$ 0<-x<\delta'$$ so $$0<y<\delta'$$ and by the definition of the limit from above taking $\delta = \delta'$ $$0 <y < \delta \implies |f(y)-L| < \epsilon \implies |f(-x) - L| < \epsilon$$ which is what we wanted to prove.

Mario Aldean
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  • But what is the point of changing the variable, it will be the same as -(-x) ? The limit from the left should be 0 - (-x) < δ if im not mistaken? – USER149372 Feb 24 '20 at 05:43
  • @USER149372 I've edited my answer, the link you provided is quite confusing and I suspect the person that answered wasn't using the same definition of limits from above or below, even though they're equivalent. – Mario Aldean Feb 24 '20 at 06:10
  • Should it also have lim x→0+f(x) is the same as lim y→0+f(y) ? – USER149372 Feb 24 '20 at 06:23
  • @USER149372 No, x = -y not x =y so lim x→0+ f(-x) is the same as lim y→0+f(y) or lim x→0+f(x) is the same as lim y→0-f(y) – Mario Aldean Feb 24 '20 at 06:28