# Thread: Proper Notation Limits...

1. ## Proper Notation Limits...

I had a quick question which might be stupid, but I figured I should ask it as I am uncertain.

Given:
$\displaystyle \lim_{x,y\to (0,0)}\frac{x^2}{x^2+y^2}$

Then switching to polar coordinates yield:
$\displaystyle \lim_{r\to 0}\frac{r^2cos^2\theta}{r^2}$

Is my notation for the upper limit correct? We want to find the limit as r -> 0 and if the limit exists it doesn't matter how we approach the origin. Therefore $\displaystyle \theta$ can be anything. And if we find that the limit is left with only
$\displaystyle \theta$ then it means the limit doesn't exist.

2. Consider two different path of approach: $\displaystyle y=0~\&~y=x$.
Do you get different limits?

3. Originally Posted by Alterah
I had a quick question which might be stupid, but I figured I should ask it as I am uncertain.

Given:
$\displaystyle \lim_{x,y\to (0,0)}\frac{x^2}{x^2+y^2}$
Then switching to polar coordinates yield:
$\displaystyle \lim_{r\to 0}\frac{r^2cos^2\theta}{r^2}$
Is my notation for the upper limit correct? We want to find the limit as r -> 0 and if the limit exists it doesn't matter how we approach the origin. Therefore $\displaystyle \theta$ can be anything. And if we find that the limit is left with only $\displaystyle \theta$ then it means the limit doesn't exist.
Looks good to me :-)

4. Yes, you do get different limits. The assignment calls for me to switch to polar coordinates however.