For my high school AP Calculus class, we were assigned the task of using implicit differentiation to find the slope of the following relation at $(7,0)$:
$y^2=(x^2-49)/(x^2+49)$
After working out that the slope does not exist, my friend in college said that further in the study of calculus, you'll find that it actually DOES exist. If the slope does exist, what is it, and how would one find it?
First note that the domain of your relation is $|x|\geq7$ because we need $\frac{x^2-49}{x^2+49}$ to be at least zero. Therefore the point $(7,0)$ is just on the "edge" of the domain.
Implicitly differentiating gives
$$2yy'=\frac{2x(x^2+49)-2x(x^2-49)}{(x^2+49)^2}=\frac{196x}{(x^2+49)^2}$$
so
$$y'=\frac{196x}{2y(x^2+49)^2}$$
Now assume $x\geq7$ (as we must, given the domain). Then as $y\to0^{-}$, we have $y'\to-\infty$. And as $y\to0^{+}$, we have $y'\to\infty$. This shows we have a vertical tangent line at $(7,0)$.
Thus the "slope" of the curve does not exist at $(7,0)$.