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This answer provides a nice way to change variables within a limit:

If the functions $g:\ A\to B$ and $f:\ B\to C$ have limits $$\lim_{x\to\xi}g(x)=:\eta\ ,\qquad \lim_{y\to\eta}f(y)=:\zeta\ ,$$ and if $f$ is continuous at $\eta$ in case $\eta$ occurs as value of $g$, then $$\lim_{x\to\xi}f\bigl(g(x)\bigr)=\lim_{y\to\eta} f(y)\ .$$

This holds also if any one of $\xi$, $\eta$, $\zeta$ is $\ =\infty$.

However, this result requires prior knowledge of the existence of limits. Suppose I wanted to prove that $$ \lim_{x \to a} \frac{f(x) - f(a)}{x - a} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ whenever either limit exists. Unfortunately, I cannot directly use the above result because (1) one limit existing does not imply the existence of the other, and (2) neither limit expression is even continuous. The above case is pretty easy to do by hand, but for other cases, it's often nontrivial or tedious. Is there a more general result that allows us to change variables under a limit without assuming existence of the resulting limit?

Frank
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  • I think since continuity of composition of continuous functions does not depend on derivatives at all, you can just use the above as a lemma in showing such a thing – Dominic Petti May 13 '21 at 05:17
  • In this particular case, where you're just changing from $x \to a$ to $h=x-a \to 0$, the equivalence of the two expressions follows immediately from the $\epsilon$-$\delta$ definition of the limits (which involves $0<|x-a|<\delta$ in one case and $0<|h-0|<\delta$ at the corresponding location in the other case). – Hans Lundmark May 13 '21 at 05:27
  • Right, I did note that proving this is pretty easy. It's just a toy example to illustrate why change of variables without knowing that the resulting limit exists might be useful. – Frank May 13 '21 at 06:50

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