Say we have a curve like this: $$ xy + y^2x^2 = x.$$ Let's call it the curve $\mathcal{C}$.
Say then that we want to find the limit of the quantity $x^2/y$ (if it exists) as $x \to - \infty$ on the curve $\mathcal{C}$. I may have designed this problem badly, but go on the assumption that we cannot solve for $y$ in terms of $x$ explicitly. (This problem is one I've made up to be similar to another problem, and in that one, that's part of the difficulty.)
How do we approach this problem? I know the formal definitions for limits, but I can't reconcile those with the fact that it must stay on the curve $\mathcal{C}$.
For every point $(x,y)$ on $C$, $(xy+\frac12)^2=x^2y^2+xy+\frac14=x+\frac14$ hence $x\geqslant-\frac14$. Thus, limits when $x\to-\infty$ are undefined on $C$.