# Conditions to partially take a limit of composition of function [closed]

Here is my question.

I have two continuous functions $f_n(x)$ and $a_n(x)$. I know for a fact that $\lim_{n\rightarrow\infty}a_n(x) = a(x)$, and I would like to perform a partial application of the limit to the composition of functions, i.e. something like

$$\lim_{n\rightarrow\infty}f_n(a_n(x)) =(?) \lim_{n\rightarrow\infty} f_n(a(x))$$

My gut feeling is that in order for this to hold, I need $f_n(x)$ to satisfy uniform convergence, which I can obtain for $f_n$ through an application of Dini's theorem as the function meets the requirements.

The reason I want to take the limit only partially is because I want to rely on regular variation properties of $f_n$ later on.

It would be great if any of you could confirm/infirm.

Thanks and cheers!

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## closed as off-topic by Ricardo Andrade, Lucia, David White, Chris Godsil, Jack HuizengaNov 26 at 3:34

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ricardo Andrade, Lucia, David White, Chris Godsil, Jack Huizenga
If this question can be reworded to fit the rules in the help center, please edit the question.

If

• $g_n \colon A \to B$ and $g \colon A \to B$,

• $f_n \colon B \to C$ are continuous,

• $f_n \to f$ uniformly on $B$, and

• $g_n \to g$ pointwise on $A$,

then for each fixed $x \in A$ we have

$$\lim_{n \to \infty} f_n(g_n(x)) = f(g(x))$$

since $g_n(x)$ is a convergent sequence in $B$ (see exercise 9 of chapter 7 in Rudin's Principles of Mathematical Analysis). The fact that

$$\lim_{n\to\infty} f_n(g(x)) = f(g(x))$$

follows from the pointwise convergence of $f_n \to f$. Your result follows.

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