4

I've read several answers here on Stackexchange about this, but the actual requirements on $f$ and $g$ are different in different answers, and sometimes in quite substantial ways. Furthermore, I think there are some subtleties here relating to the domains of $f$ and $g$, especially when talking about $g^{-1}$.

I'll begin by giving a motivating example. Suppose we wish to find: $$ \lim_{t\to 0}\frac{\sin(\tan t)}{\tan t}. $$

What we do is make the substitution $x = \tan t$ and then use that $$ \lim_{x\to 0}\frac{\sin(x)}{x} = 1. $$

So what we have is an inner function $g(t) = \tan t$ and an outer function $f(x) = \frac{\sin x}{x}$. We're making the claim that if $\lim_{t\to a} g(t) = L$, then $$ \lim_{t \to a} f(g(t)) = \lim_{g(t) \to L} f(g(t)) $$

My question is about the assumptions on $f$ and $g$ for this to hold. I know that the limit of $g$ at $a$ has to exist. But what about other requirements? I've seen continuity for $f$ be demanded, but also versions where $f$ has no requirements whatsoever, but $g$ has to be injective and continuous. Other claims have been that $g$ simply has to be invertible. In particular, for this example $f$ is not continuous at the point of interest.

Peatherfed
  • 726
  • 2
    How do you define the meaning of$$\lim_{x\to a}f(x)=L?$$ – José Carlos Santos Jun 30 '21 at 21:56
  • I use the standard $\varepsilon$-$\delta$ definition. If it matters, although I think it's clear from context, I'm talking about strictly real functions of one variable here. – Peatherfed Jun 30 '21 at 22:03
  • "I've read several answers here on Stackexchange about this, but the actual requirements on $f$ and $g$ are different in different answers, and sometimes in quite substantial ways." OK, can you cite several answers here on MSE that are confusing you in this regard? – zhw. Jul 01 '21 at 00:19
  • This: https://math.stackexchange.com/questions/167926/formal-basis-for-variable-substitution-in-limits and this: https://math.stackexchange.com/questions/1069642/finding-a-limit-using-change-of-variable-how-come-it-works are two examples. – Peatherfed Jul 01 '21 at 00:29
  • Oh, and here's another related document, not from stackexchange though: https://www3.nd.edu/~math/restricted/CourseArchive/100Level/166/1662000S/Misc/Substitutions.pdf Here the author requires the function $g$ to be non-constant, but I think he only does that because he wants invertibility. – Peatherfed Jul 01 '21 at 00:40
  • There are unfortunately several “standard” definitions in use. Some sources require $f$ to be defined in a punctured interval around $a$, others only require $a$ to be an accumulation point of the domain of $f$. Some use $0<|x-a|<\delta$, others $|x-a|<\delta$. The answer to your question depends very much on these details, so you will need to be more specific. – Hans Lundmark Jul 01 '21 at 05:46
  • That the function you will use to change the variable exists. –  Jun 30 '21 at 22:03
  • I see, then I'll be more specific. Let $f$ be defined in a punctured, open interval around $a$. Then if for all $\varepsilon > 0$ there exists a $\delta > 0$ such that $|f(x) - L| < \varepsilon$ whenever $0 < |x-a| < \delta$, then we say that the limit... and so on. – Peatherfed Jul 01 '21 at 13:05

0 Answers0