It doesn't seem necessary to go to so much trouble, does it I don't know where this answer is provided, and it feels strange (as if you wrote the "+"between the two limits as a multiplication symbol).

I want to know whether the condition f (0,0) = 0 is missing in your topic. If not, it should be simple.

Let the function f(x, y) satisfy f (0,0) ≠ 0. At this time, because of continuity, lim[x→0, y→0] f(x, y)≠0, so the two limits in c and d must be infinite, so they do not exist.

If there is a condition of f (0,0) = 0 in the topic, it is a bit troublesome. The integral limit is omitted below.

lim f(x，y)/[|x|+|y|]

=lim [f(x，y)-f(0，0)]/[|x|+|y|]

=lim (Ax+By)/[|x|+|y|]

Let (x, y) tend to (0,0) along the positive and negative directions of the coordinate axis, and we can conclude that the limit does not exist.

For the second similarity, let (x, y) follow y = x? Tending to (0,0), we can get different limits.

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