Never mind, you got the answer right.
And in your other case
Now remember the logarithm rule ?
That means you can take the power out as a factor...
So it becomes
Notice though that you can't solve this using direct substitution, since is undefined. It will tend to the indeterminate . This one requires a bit more thought.
The crucial point, for both of these, is that once you have put the problem in polar coordinates, the limit, as r goes to 0, does not depend on ! Since r itself determines "closeness" to (0,0), that is what you must have in order that you do not get different limits approaching (0,0) along different paths- in order that the limit exist.