Just want to make sure I'm doing this right.

$$\lim_{(x,y)-(0,0)} \frac{x^{3}+xy^{2}}{x^{2}+y^{2}}$$
If I set y to 0, this boils down to x, and the limit of x as x goes to 0 is 0.
If I set x to 0, then this boils down to $$\frac{0}{y^{2}}$$ or 0. And since I get the same limit either way, my limit is 0.

Question

Just want to make sure I'm doing this right.

$$\lim_{(x,y)->(0,0)} \frac{x^{3}+xy^{2}}{x^{2}+y^{2}}$$
If I set y to 0, this boils down to x, and the limit of x as x goes to 0 is 0.
If I set x to 0, then this boils down to $$\frac{0}{y^{2}}$$ or 0.  And since I get the same limit either way, my limit is 0.

ash2326 · Answer

Yeah , you're right:)

bahrom7893 · Answer

yes

chriss · Answer

Awesome, and one more question.  If for a given expression, I am trying to prove the limit does not exist, then I would use this same method, and if I get different limits then I have shown that the limit does not exist.

ash2326 · Answer

yeah:D

chriss · Answer

Great.  Thank you very much :)

ash2326 · Answer

Welcome:D

anonymous · Answer

I love this type of question all I have to say yes or no and get medals! :D

anonymous · Answer

yes, no
ok i have a question 
if x and y are not both zero, why isn't this just 
$$x$$?

anonymous · Answer

and so while the limit is in fact zero, i don't think one computes a limit of this type by setting one variable = 0 and then taking the limit as the other goes to zero.
that is only going to zero from one direction

chriss · Answer

right because the next problem is giving me an issue... I basically just got the same result as in my first example but I know the limit is not supposed to exist here...

$$\lim_{(x,y)->(2,0)} \frac{xy-2y}{x^{2}+y^2}$$