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Math Help - limit definition

  1. #1
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    limit definition

    Let f(x,y)=sin|x|cos|y|.

    (a) Show from the limit definition that f_y(0,0) exists, and find its value.

    (b) Show from the limit definition that f_y(0,0) does not exist.

    My Attempt:

    (a)
     \lim_{\Delta y \to 0} \frac{f(x_0, y_0 + \Delta y)-f(x_0,y_0)}{\Delta x}

     \lim_{\Delta y \to 0} \frac{sin|0+|cos|0+\Delta x|-sin|0|xos|0|}{\Delta y}

    \lim_{\Delta y \to 0} \frac{0}{\Delta y} = 0

    Is the value zero?

    (b)
    I can't prove that the limit doesn't exits because it turns out it exists:

     \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x, y_0)-f(x_0,y_0)}{\Delta x}

    \lim_{\Delta x \to 0} \frac{sin|0+ \Delta x|cos|0|-sin|0|cos|0|}{\Delta x}

    \lim_{\Delta x \to 0} \frac{sin|\Delta x|}{\Delta x} = 1

    (since limit as x ->0 sinx/x=1)

    I'm confused can anyone help?
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  2. #2
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    Quote Originally Posted by demode View Post
    Let f(x,y)=sin|x|cos|y|.

    (a) Show from the limit definition that f_y(0,0) exists, and find its value.

    (b) Show from the limit definition that f_y(0,0) does not exist.

    My Attempt:

    (a)
     \lim_{\Delta y \to 0} \frac{f(x_0, y_0 + \Delta y)-f(x_0,y_0)}{\Delta x}

     \lim_{\Delta y \to 0} \frac{sin|0+|cos|0+\Delta x|-sin|0|xos|0|}{\Delta y}

    \lim_{\Delta y \to 0} \frac{0}{\Delta y} = 0

    Is the value zero?
    Yes, that's correct.

    (b)
    I can't prove that the limit doesn't exits because it turns out it exists:

     \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x, y_0)-f(x_0,y_0)}{\Delta x}

    \lim_{\Delta x \to 0} \frac{sin|0+ \Delta x|cos|0|-sin|0|cos|0|}{\Delta x}

    \lim_{\Delta x \to 0} \frac{sin|\Delta x|}{\Delta x} = 1

    (since limit as x ->0 sinx/x=1)

    I'm confused can anyone help?
    But you do not have \lim_{x\to 0}\frac{sin x}{x}, you have \lim_{x\to 0}\frac{sin|x|}{x}

    Look at the two one sided limits, as x goes to 0 from above and as x goes to 0 from below, remembering that for x< 0, sin(|x|)= sin(-x)= - sin(x).
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  3. #3
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    But you do not have \lim_{x\to 0}\frac{sin x}{x}, you have \lim_{x\to 0}\frac{sin|x|}{x}

    Look at the two one sided limits, as x goes to 0 from above and as x goes to 0 from below, remembering that for x< 0, sin(|x|)= sin(-x)= - sin(x).
    Could you explain a little bit more please because I'm not sure if I really understand this. We are not concerned with x<0, we only care about (0,0), and as x goes to 0, sin|0|=0 and that makes the limit zero!
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