I’m wondering if uniform convergence can hold if we have a $0/0$ indeterminate form for certain values of $x$ for the limit function.
More specifically, let $f_n$ and $g_n$ be two sequences of functions that are each uniformly convergent to functions $f$ and $g$, respectively, where $g(x)=0$ whenever $f(x)=0$ such that $lim_{t\rightarrow x}{f(t)/g(t)}$ exists and is nonzero. Is $f_n/g_n$ uniformly convergent to $f/g$ in a set containing any zero of $f$ and $g$?
I understand that there are concerns of removable discontinuities here, but my thought is that this would not affect convergence. I have tried researching uniform convergence and have seen no mention of $0/0$ scenarios so feel like the result is not true.
You have to be more specific with your set: consider the set $[1,\infty)$ and the functions $$f_n(x) = \frac{1}{x+n} \quad g_n(x) = \frac{1}{x+2n}.$$ It is clear that $f_n, g_n \to 0$, but $\lim_{n \to \infty} \frac{f_n(x)}{g_n(x)} = 2.$
However, it is clear that this convergence can never be uniform.
Edit 1: consider $$f_n(x) = \frac{1}{x+n} + (x-1) \quad g_n(x) = \frac{1}{x+2n} + (x-1).$$
It is clear that $\lim_{t \to x}\frac{f(x)}{g(x)} = 1$ for all $x\geq 1.$
Both $f_n(x),g_n(x) \to x-1$ uniformly. Note that $$\frac{f_n(x)}{g_n(x)} = \frac{\frac{x+2n}{x+n}+(x-1)(x+2n)}{1+(x-1)(x+2n)}.$$
When $x \neq 1$, $\lim_{n \to \infty}{f_n(x)}{g_n(x)} = 1$.
However, when $x = 1$, $\lim_{n \to \infty}{f_n(1)}{g_n(1)}=2.$
Does this meet your criterion?