I'd be interested to know when, if $f(x,y)=g(x,y)+O(x^n)$ we have that $\lim_{y\rightarrow c}=\lim_{y\rightarrow c}g(x,y)+O(x^n).$ Are there conditions of $f$ and/or $g$ that make sure that this is satisfied?
In particular, assume that we have a function $f(x,y)$ and that we Taylor expand it as a function of $x$ around $a$: $f(a+x,y)=f(a,y)+\frac{\partial f}{\partial x}(a,y)\cdot x+\frac{\partial^2 f}{\partial x^2}(a,y)\cdot \frac{x^2}{2}+\cdots+O(x^n).$
I wish to study the behaviour of $f(x,y)$ as $y$ tends to a value on the boundary of the domain of $f(x,y)$, say $y\rightarrow c$. I therefore wonder under what conditions on $f$ $\lim_{y\rightarrow c}f(a+x,y)=\lim_{y\rightarrow c}f(a,y)+\lim_{y\rightarrow c}\frac{\partial f}{\partial x}(a,y)\cdot x+\lim_{y\rightarrow c}\frac{\partial^2 f}{\partial x^2}(a,y)\cdot \frac{x^2}{2}+\cdots+O(x^n).$ In other words, when is the $O(x^n)$ term still $O(x^n)$ as $y\rightarrow c$?
I've been hoping to find some smoothness constraints on $f$ that would guarantee that the expansion still is $O(x^n)$, but so far I haven't been able to find any.