What does it mean for one function to be the limit of another

Question

Consider two real valued functions $F(a,x)=\frac{1}{a}+x^2$ and $G(x)=x^2$.

It is clear that as $a$ tends to infinity, $F$ tends to $G$; but what does that mean?

Shouldn't there be some kind of metric $M$, to measure the distance between $F$ and $G$ and to say that as a increases $M$ decreases and that there is no positive number that is the minimum value $M$ takes?

What kinds of metrics are there for problems like this?

Answer 1

$F$ tends to $G$ not only point-wise but uniformly.

Obviously, for any $\varepsilon>0$ there exists an $a$ such that

$$\sup_x |F(a,x)-G(x)|=\sup_x\left|\frac1a+x^2-x^2\right|=\frac1a<\varepsilon.$$

In this respect the measure of closeness is $\sup_x|F-G|$.

Answer 2

There is indeed a geometric picture underlying this kind of convergence, specifically a topology. Specifically, the topology corresponding to pointwise convergence is the point-open topology: namely, the topology generated by sets of the form $$\{f: f(x)\in U\}$$ for $x\in\mathbb{R}$ and $U\subseteq\mathbb{R}$ open.

(As zoli says, the limit you mention is in fact an instance of a stronger form of convergence. There are lots of examples of pointwise but not uniform convergence: e.g. ${x\over a}+x^2\rightarrow x^2$ pointwise, but not uniformly, as $a\rightarrow\infty$.)