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Math Help - prove the limit exist (Partial Differential)

  1. #1
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    prove the limit exist (Partial Differential)

    Hi everyone this is my first time here - I'm completely confused not so I thought I may as well ask complete stangers for help!

    the question is..
    Determine whether the following limit exists. If so, find its value.

    lim [sin(x^2+y^2+z^2)] / (x^2+y^2+z^2)^1/2
    x,y,z-->(0,0,0)

    first, how can i prove that the limit is exist?

    i'm thinking of to get the above limit to be something like
    [sin (x)]/x = 1
    or..
    should i use quotient rule..?
    hope that someone can guide me solving this question..
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  2. #2
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    Quote Originally Posted by nameck View Post
    Hi everyone this is my first time here - I'm completely confused not so I thought I may as well ask complete stangers for help!

    the question is..
    Determine whether the following limit exists. If so, find its value.

    lim [sin(x^2+y^2+z^2)] / (x^2+y^2+z^2)^1/2
    x,y,z-->(0,0,0)

    first, how can i prove that the limit is exist?

    i'm thinking of to get the above limit to be something like
    [sin (x)]/x = 1
    or..
    should i use quotient rule..?
    hope that someone can guide me solving this question..
    Yes, use spherical polar coords..

     <br />
x = \rho \cos \theta \sin \phi,\;\;<br />
y = \rho \sin \theta \sin \phi<br />
,\;\;<br />
z = \rho \cos \phi<br />

    so x^2+y^2+z^2 = \rho^2
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  3. #3
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    i'm sorry..
    i never learn that kind of equation..
    got other ideas to solve it?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nameck View Post
    Hi everyone this is my first time here - I'm completely confused not so I thought I may as well ask complete stangers for help!

    the question is..
    Determine whether the following limit exists. If so, find its value.

    lim [sin(x^2+y^2+z^2)] / (x^2+y^2+z^2)^1/2
    x,y,z-->(0,0,0)

    first, how can i prove that the limit is exist?

    i'm thinking of to get the above limit to be something like
    [sin (x)]/x = 1
    or..
    should i use quotient rule..?
    hope that someone can guide me solving this question..
    Is this \lim_{(x,y,z)\to(0,0,0)}\frac{\sin\left(x^2+y^2+z^  2\right)}{\left(x^2+y^2+z^2\right)^{\frac{1}{2}}} or \lim_{(x,y,z)\to(0,0,0)}\left(\frac{\sin\left(x^2+  y^2+z^2\right)}{x^2+y^2+z^2}\right)^{\frac{1}{2}}
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  5. #5
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    Quote Originally Posted by nameck View Post
    i'm sorry..
    i never learn that kind of equation..
    got other ideas to solve it?
    How about

    0 \le \sin \left(x^2+y^2+z^2\right) \le x^2+y^2+z^2

    so

     <br />
0 \le \frac{\sin \left(x^2+y^2+z^2\right)}{\sqrt{x^2+y^2+z^2}} \le \sqrt{x^2+y^2+z^2}<br />

    Now letting (x,y,z) \to (0,0,0) gives your answer.
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  6. #6
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    ok this is my idea..
    if i let t=x^2+y^2+z^2, the limit should be like..

    lim (sin t) / (t^1/2)

    apply L'Hospital rule,

    (cos t) / (t^-1/2) = t^1/2 cos t

    am i right?

    t=0.. so...

    (cos t) / (t^-1/2) = t^1/2 cos t
    = 0^1/2 cos 0
    = 0
    ok.. how am i going to proceed?
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  7. #7
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    so.. is my calculations above is correct?
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