Asymptotic behaviour of implicit functions

Question

Suppose we have an implicit equation $F\left(x,y\right)=0$ which we know defines $y = y(x)$ as a function of $x$. Are there sufficient or necessary conditions under which we can obtain information about the asymptotic behaviour of $y(x)$? Particularly, what is the limit $\lim _{x\to +\infty }y\left(x\right)$ (if it even exists)? Or if we constrain ourselves to the domain $x,y>0$ then what is $\lim _{x\to 0^+ }y\left(x\right)?$

Answer 1

You can't say anything in general. The implicit function theorem only guarantees that for every $(x_0, y_0)$ such that $F(x_0, y_0) = 0$ and $\partial_yF(x_0, y_0)\neq 0$, there is a (possibly small) neighborhood of $x_0$ where $y$ is defined as an implicit function of $x$. That neighborhood could be a very small interval, so $y(x)$ may not be defined for arbitrarily large values of $x$.

Simple example: $F(x,y) = x^2 + y^2 -1$ is the circle. There, $x$ can only go from $-1$ to $1$. You can't take the limit when $x\rightarrow \infty$. So first, you should assume that the $x$ coordinate is unbounded in the set $F(x,y) = 0$. But that may not be enough because the implicit function theorem is still local, and only gives local information (i.e., near the point $x_0$).