f(x,y) = 1/(x2 - y)
(a) For critical points, we need to find the points where both partial derivatives equal zero. So, using chain rule,
δf/δx = -1 / (x2 - y)2 * 2x
Setting this equal to zero, we find x = 0
δf/δy = -1 / (x2 - y)2 * (-1)
Setting this equal to zero, there are no solutions. Therefore, this function has no critical points. We would need to find a point where both partial derivatives are simultaneously zero to have a critical point.
(b) Conceptually, a limit of a multivariable function existing means we can approach that point from any direction (or path) and get the same limit. In this case, consider simply the line x = 0 and decide for yourself whether the limit is the same as y approaches 0 from the positive and negative side. Demonstrating that the limit from two different directions (or along two different paths) is different is sufficient to prove a limit doesn't exist, just as it is for single variable functions.
If you need clarification on that, feel free to let me know.