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Math Help - continuity

  1. #1
    mms
    mms is offline
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    continuity

    show that the following function is continuos at (0,0)

    <br />
f(x,y) = \left\{ \begin{array}{l}<br />
\frac{{xsen(x)}}{{\sqrt {x^2 + y^2 } }}\,\,\,if(x,y) \ne (0,0) \\ <br />
0\,\,\,if\,(x,y) = (0,0) \\ <br />
\end{array} \right.<br /> <br />

    i need to know how to do this kind of problems, with this example i think i may be able to do other problems of the same kind

    thanks
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  2. #2
    MHF Contributor chisigma's Avatar
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    To aswering the question it is necessary to valuate the...

    \lim_{(x,y) \rightarrow (0,0)} \frac{x\cdot \sin x}{\sqrt{x^{2} + y^{2}}} (1)

    At this scope we use polar coordinates...

    x= \rho\cdot \cos \theta

    y= \rho\cdot \sin \theta (2)

    ... and limit (1) becomes...

    \lim_{\rho \rightarrow 0} \cos \theta\cdot \sin (\rho\cdot \cos \theta) (3)

    Bur the limit (3) is 0 independently from \theta so that the function You have proposed is continous in (0,0)...

    Kind regards

    \chi \sigma
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