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Math Help - Multivariable limit.

  1. #1
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    Multivariable limit.

    Hi,
    The question asks to check if this funcion: sin( x^2 + y^2 ) / x^2 + y^2 is continuous or not which I already found out since its domain is {(x,y): x!=0 and y!=0}
    But now its asks to check whether it can be redefined to be continous or not about which I am very confused. So here's what I think:
    The function will become one when z tends to zero whihc makes the function continuous at zero and I think everywhere. But I don't have the clear understanding of this thing.
    Kindly clear my understanding about it up since I am having a limited knowldge in this area and let me know if what I think is correct or not.
    Thank you so much!
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  2. #2
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    Quote Originally Posted by Sonia View Post
    But now its asks to check whether it can be redefined to be continous or not about which I am very confused.
    Have you tried to substitute \displaystyle x = r\cos \theta, y = r\sin \theta ?
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    In that case, the possible answer would be one. But from my understanding, arn't polar coordinates used to check for limit values?
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    The only possible discontinuity is \displaystyle (x, y) = (0, 0).

    So you need to see if \displaystyle \lim_{(x, y) \to (0, 0)}\frac{\sin{(x^2 + y^2)}}{x^2 + y^2} exists. As was suggested, this is easiest to evaluate if you convert to polars.
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  5. #5
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    Quote Originally Posted by Sonia View Post
    In that case, the possible answer would be one. But from my understanding, arn't polar coordinates used to check for limit values?
    Polar coordinates are used for many purposes! But, specifically, if you have a two dimensional problem and are taking the limit at (0, 0), then polar coordinates have the advantage that the distance from a point to (0, 0) is measured by the single variable r. If taking the limit as r goes to 0 gives a value that does NOT depend on \theta, then that value is the limit as we approach (0, 0) from any direction. If the result does depend on \theta, then the limit does not exist.
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