(i) Yes. Simplify .
Now compute the limit by converting to polar coordinates.
This tells us
so we can define to make the function continuous at the origin.
Alternatively, we have
and
Now,
so by the squeeze theorem,
and approaches 1 as we approach the origin.
(ii) No. Expand the fraction.
and are undefined, so there is no way to make continuous at (0, 0).
(iii) No. Similarly,
is undefined when .