Question

Can someone explain why this limit doesn't exist?

Answer 1

For a limit to exit, it must exist regardless of the direction from which you approach the limit point. Notice that if you go to the x-axis, where y = 0, the function becomes
\[ f(x,0) = \frac{x^4}{x^2} \]
so the limit as you approach the origin along the x axis, the limit is equal to zero.

However, if you sit on the y-axis, where x = 0, the function becomes
\[f(0,y) = -2\frac{y^2}{y^2} \]
so as you approach the origin along the y axis, the limit is equal to -2.

Because you find different answers for different directions, the two-dimensional limit as (x,y) approaches (0,0) does not exist.

Answer 2

Because the function may not be continuous at a point that's the basic idea.

Answer 3

So will it always hold true, that if the limit as x approaches zero is different than the limit as y approaches zero, the limit doesn't exist?

Answer 4

Strictly speaking, whether the function is continuous at a point or not is not the issue, because limits don't care about the behavior of the function at any particular point but rather the behavior of the function as you approach those points. You could certainly have a removable discontinuity at a point and still have a valid limit at that point.


And yes, to answer your question -- if those two limits are different, the limit doesn't exist. It is an extension of your 1D calculus knowledge, where you learned that if the left-hand limit and the right-hand limit are different, then the limit doesn't exist -- but now, you can approach in ANY direction, not just from left or right.

Answer 5

It's also important to notice that the limit along the x-axis and the limit along the y-axis may be the same, but the limit as you approach along the line x = y may be different, in which case the limit STILL doesn't exist.

Answer 6

Thank you very much @Jemurray3 , that was extremely well explained :)