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Math Help - Limits

  1. #1
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    Limits

    Find the following limit if they exist, otherwise explain why they do not exist:

    \lim \frac{Re(z)}{z}
    with  z\rightarrow 0

    Since  Re(z) = x

    So \lim \frac {x}{x+iy} ??
    with  z\rightarrow 0

    How do i continue from here?
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  2. #2
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    Quote Originally Posted by Richmond View Post
    Find the following limit if they exist, otherwise explain why they do not exist:
    \lim \frac{Re(z)}{z} with  z\rightarrow 0

    Since  Re(z) = x
    So \lim \frac {x}{x+iy} ??
    with  z\rightarrow 0
    How do i continue from here?
    Consider the limit along the path x=y. What is it?

    Consider the limit along the path y=0. What is it?

    What does that tell you?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Consider the limit along the path x=y. What is it?

    Consider the limit along the path y=0. What is it?

    What does that tell you?
    I'm not pretty sure but I would guess that the limit along the path x = y is 0 and the limit along the path y = 0 is 0 as well.

    Thus the limit of this function does not exist?
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  4. #4
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    Quote Originally Posted by Richmond View Post
    I'm not pretty sure but I would guess that the limit along the path x = y is 0 and the limit along the path y = 0 is 0 as well.

    Thus the limit of this function does not exist?
    z-->0 implies (x,y)-->(0,0)

     \lim_{(x,y) \to (0,0)} f(x,x) = \lim_{(x,y) \to (0, 0)} \frac {x}{x+ix} = \frac {1}{1+i}

     \lim_{(x,y) \to (0,0)} f(x,0) = \lim_{(x,y) \to (0, 0)} \frac{x}{x} = 1


    Since different paths to the origin give different limits, the limit does NOT exist
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