# Proper Notation Limits...

• Sep 19th 2009, 02:24 PM
Alterah
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.

• Sep 19th 2009, 02:58 PM
Plato
Consider two different path of approach: $\displaystyle y=0~\&~y=x$.
Do you get different limits?
• Sep 19th 2009, 02:59 PM
Jester
Quote:

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 :-)
• Sep 19th 2009, 06:10 PM
Alterah
Yes, you do get different limits. The assignment calls for me to switch to polar coordinates however.