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Math Help - limit

  1. #1
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    limit

    If  f(x,y) = \begin{cases} x \sin \frac{1}{y} \ \ \text{if} \ y \neq 0 \\ 0 \ \ \ \ \ \ \ \ \ \text{if} \ y = 0 \end{cases} , show that  \lim_{(x,y) \to (0,0)} f(x,y) = 0 but that  \lim_{x \to 0} \left(\lim_{y \to 0} f(x,y) \right) \neq \lim_{y \to 0} \left(\lim_{x \to 0} f(x,y) \right).

    So for the first part, I simply plugged in  (x,y) = (0,0) and got  0 . For the second part, I think the limits oscillate and don't exist?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by heathrowjohnny View Post
    If  f(x,y) = \begin{cases} x \sin \frac{1}{y} \ \ \text{if} \ y \neq 0 \\ 0 \ \ \ \ \ \ \ \ \ \text{if} \ y = 0 \end{cases} , show that  \lim_{(x,y) \to (0,0)} f(x,y) = 0 but that  \lim_{x \to 0} \left(\lim_{y \to 0} f(x,y) \right) \neq \lim_{y \to 0} \left(\lim_{x \to 0} f(x,y) \right).

    So for the first part, I simply plugged in  (x,y) = (0,0) and got  0 . For the second part, I think the limits oscillate and don't exist?
     \lim_{y \to 0} \left(\lim_{x \to 0} f(x,y) \right)=0

    as:

    \lim_{x \to 0} f(x,y)=0

    The problem is with the other limit as for any x\ne 0:

    \lim_{y \to 0} f(x,y)

    does not exist

    RonL
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