Proof that we are allowed to make substitutions to evaluate limits?

Question

Basically, if we want $\lim_{x\to x_0}f(x)$, why are we 'allowed' to instead find the limit

$\lim_{u\to u_0}f(u)$ where $x = k(u)$, where $k(u)$ is a function in $u$ such that $\lim_{u\to u_0}k(u)= x_0$.

Sorry if this isn't precise enough.. essentially, I want to know why, for example, when we find the limit

$\lim_{x\to 0}f(x)$ we're allowed to instead find the limit

$\lim_{2t\to 0}f(t)$ with $x = 2t$.

I think it's trivial because it's basically 'the same limit', but I'd like to verify that intuition.

Please improve your MatJax formatting, it is hard to read. E.g. use \lim_x for $\lim_x$, \to for $\to$, etc. — M. Winter, Nov 15 '17 at 17:11
Maybe Formal basis for variable substitution in limits helps? — Sil, Nov 15 '17 at 17:11
It's a standard procedure and it should be available in textbooks in the same manner as sum/product/quotient rule of limits are available. The precise statement is: if $\lim_{x\to a} f(x) =L$ and $\lim_{t\to b} g(t) =a$ and further $g(t) \neq a$ as $t\to b$ then $\lim_{t\to b} f(g(t)) =L$. — Paramanand Singh, Nov 15 '17 at 17:35
@ParamanandSingh What exactly do you mean by the last line? The answer to the other question just requires f to be continuous at a. — John, Nov 15 '17 at 17:58
The function f must be continuous limits must exist (not be infinity). This is a standard theorem and it should be proven via delta epsilon proofs. But you are absolutely correct that it should not be taken for granted. — fleablood, Nov 15 '17 at 18:38
@Saad: it means that there is a deleted neighborhood $I$ of $b$ such that $g(t) \neq a$ for all $t\in I$. — Paramanand Singh, Nov 15 '17 at 21:47

score 1 · Answer 1 · answered Nov 15 '17 at 19:02

1

A simple version of the substitution rule: Suppose

i) $f$ is defined in a deleted neighborhoood of $a;$

ii) $g$ is a strictly increasing (or strictly decreasing) continuous function defined in a neighborhood of $b,$ with $g(b) = a.$

Then

$$\lim_{x\to a} f(x) = L\, \iff \,\lim_{y\to b} f\circ g(y) = L.$$

I'll omit the proof here, except to say the conditions on $g$ imply that we can move back and forth from $a$ to $b$ in the most natural way, using the maps $g$ and $g^{-1}.$

answered Nov 15 '17 at 19:02

zhw.

105,693

This is different from what the other question has, which is different from what someone in the comments said.. And I don't think the 3 ways of formulating it are equivalent either. Which one would be the most general? I'd guess the one in the other question since we don't need g to be strictly increasing or decreasing (or even continuous) there.. – John Nov 15 '17 at 19:55
@Saad This is the simplest one to use, easiest to understand, and the one you will see most often - all IMHO. – zhw. Nov 15 '17 at 21:06
+1 for framing the conditions in a simple manner. @Saad : you may look at the following answer where I describe the substitution of limits https://math.stackexchange.com/a/1073047/72031 – Paramanand Singh Nov 16 '17 at 05:58

Proof that we are allowed to make substitutions to evaluate limits?

1 Answers1