Finding critical points and limits.

Question

Let f(x,y) = 1/(x² - y)

(a) Determine the critical points of f.
(b) Does the limit of the function f at the point (0, 0) exist? Justify your answer

Answer 1

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.