Show limit does not exist

Let D = R^2 / {(0.0)} and f : D > R be given by

f(x; y) = 1 - cos(x^2 - y^2)/(x^2 + y^2)^2 :

Show that lim f(x; y) does not exist.

If you approach along the x axis (y=0), you get
Lim (as x approaches 0) of
[ 1 - cos(x^2)] / (x^2)^2 = x^4

Since this is 0/0
use the derivative/derivative rule (L'Hôpital's Rule)

d(1 - cos(x^2))/dx = 2xsin(x^2)
divided by
d(x^4)/dx = 4x^3

IF the limit exists for the original function, then it must be equal to the limit of the derivative/derivative
= limit of 2x*sin(x^2) / 4x^3

Use the fact that as x approaches 0, sin(x) = x
this allows you to replace sin(x^2) by x^2
and your get

limit = 2x^3 / 4x^3 = 1/2

So IF the limit exists, then it is equal to 1/2

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Approach from a diagonal (x = y)
Limit, as x approaches 0 of
[1 - cos(x^2 - x^2)] / (x^2 + x^2)^2
[1 - cos(0)] / (2x^2)^2
0 / 4x^4

This limit, IF it exists, will be equal to 0, which is clearly NOT the same as 1/2

A limit (at a single point) MUST be unique. Since this one is not unique, it is not a limit.

The limit does not exist.

is everything after cos inside the cos? or just one part?