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Math Help - Proper Notation Limits...

  1. #1
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    Proper Notation Limits...

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

    Given:
    \lim_{x,y\to (0,0)}\frac{x^2}{x^2+y^2}<br />

    Then switching to polar coordinates yield:
    \lim_{r\to 0}\frac{r^2cos^2\theta}{r^2}<br />

    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 \theta can be anything. And if we find that the limit is left with only
    \theta then it means the limit doesn't exist.

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  2. #2
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    Consider two different path of approach: y=0~\&~y=x.
    Do you get different limits?
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  3. #3
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    Quote Originally Posted by Alterah View Post
    I had a quick question which might be stupid, but I figured I should ask it as I am uncertain.

    Given:
    \lim_{x,y\to (0,0)}\frac{x^2}{x^2+y^2}
    Then switching to polar coordinates yield:
    \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 \theta can be anything. And if we find that the limit is left with only \theta then it means the limit doesn't exist.
    Looks good to me :-)
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  4. #4
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    Yes, you do get different limits. The assignment calls for me to switch to polar coordinates however.
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