The most promising way to find such types of limits is working in polar coordinates with the substitution...

(1)

... and searching [if it exists...] the limit with that is independent from .

Let's start with your example nr. 2. With substitutions (1) we have...

(2)

For 'small' value of is...

(3)

... so that...

(4)

And now your example nr. 1. With substitutions (1) we have...

(5)

For 'small' value of is...

(6)

... so that...

(7)

Now we observe that the limit (7) does depend from so that we must conclude that limit (5) doesn't exist...

Kind regards