What is the meaning of $f(x) \rightarrow a$ as $g(x) \rightarrow b$? The motivating example was the case:
$$f(x, y)\rightarrow0\mathrm{\ \ as\ \ }\sqrt{x^2+y^2}\rightarrow\infty$$
What exactly does this mean? I might define it as:

Any sequence $x_n$ with $g(x_n)\rightarrow b$ verifies $f(x_n)\rightarrow a$.

Is this right? Are there other definitions?

This is a bit more of a special case than that.  Consider rewriting in vector notation: we take $f:\mathbb{R}^2\rightarrow\mathbb{R}$, denoted as $f(\vec{x})$ where $\vec{x}=(x,y)$. Then $\sqrt{x^2+y^2}=\|\vec{x}\|$. So, this statement becomes
$$
f(\vec{x})\rightarrow0\text{ as }\|\vec{x}\|\rightarrow\infty,
$$
which seems a bit more intuitive. 
Formally, what this means is that for any $\epsilon>0$, there exists $M$ such that whenever $\|\vec{x}\|>M$, we have $\lvert f(\vec{x})-0\lvert<\epsilon$. In other words, we can make $f(\vec{x})$ as close to 0 as we like by choosing $\vec{x}$ sufficiently far away from the origin, in any direction.
