Limit for a two variables function Question

[Freydulf]

Hi!

I have some doubts bound up with an exercise of limits. It demands to calculate the existence of the limit in (0,0) of these two functions.

http://www.rinconmatematico.com/latexrender/pictures/bd1b868f7e07af600ea0c82b34a38137.png

http://www.rinconmatematico.com/latexrender/pictures/5a3966109b20a098800453a99467592c.png

My solution for the first one til now is:

http://www.rinconmatematico.com/latexrender/pictures/667c03805a5b4b203406f69cf574c56e.png

Well, after the substitution we have http://www.rinconmatematico.com/latexrender/pictures/04dd2173792059a5b0e5447b852baf28.png

So we can use http://www.rinconmatematico.com/latexrender/pictures/722e499f2e8ad1b9b66060f151689378.png

http://www.rinconmatematico.com/latexrender/pictures/2bde7e5f3e31f640f734b26dcf229dc4.png

Using http://www.rinconmatematico.com/latexrender/pictures/03d4bbee51cd042a5e43f4b8eed0e81e.png

http://www.rinconmatematico.com/latexrender/pictures/a4abe4b98d7e1c4782faca7c9db897f9.png

And http://www.rinconmatematico.com/latexrender/pictures/597c072a27536c4904c39673dc916ace.png

http://www.rinconmatematico.com/latexrender/pictures/b04c33391868c0ff16d9fa2933c6466b.png

Well, I don't know if I've done correctly all the process, but if I did, I guess that in the next step I have to calculate the limit by the definition to conclude the exercise, and that's one of the things that I don't know how to do.
The second exercise is similar to the first one but I don't know how to do it either. Maybe the minus changes the result and there's not limit and any necessity of employing the definition.
Anyhow, can someone lend me a hand? :)

Thanks!

 

[HallsofIvy]

For problems like this, after you have tried taking the limit along different paths and seen that you don't get different answer (which would have proved the limit does not exist), I recommend converting to polar coordinates. That way "closeness" to (0,0) is measured only by r, not the angle [/itex]\theta[/itex]. If the limit, as r goes to 0, is a constant, not depending on \theta that is the limit of the function. If the limit as r goes to 0 depends on \theta, the limit does not exist.