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Math Help - multivariable limit

  1. #1
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    multivariable limit

    not too sure how to do limits in multivariable yet.

    heres my question?:

    Limit as (x,y)-->(0,0) of (e^(xy)-1)/y.

    I know the answer is zero from the back of the book but how i did it it got negative infinite.

    I multiplied by x/x so the function was (xe^(xy)-x)/(xy) and split it into two limits. The first limit (xe^(xy))/xy goes to 0 i think, and then -x/(xy) i think goes to - infinity.

    how do i actually do this/whats wrong with what i did?
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    yes, the limit is zero, but it's enough to remember a little fact from single variable calculus: \dfrac{e^x-1}x\xrightarrow[x\to0]{}1 so write \displaystyle\frac{{{e}^{xy}}-1}{y}=x\cdot \frac{{{e}^{xy}}-1}{xy}, and the rest follows.
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  3. #3
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    oh! right! thank, that was much more simple than i was making it out to be
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  4. #4
    MHF Contributor chisigma's Avatar
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    It is false that...

    \displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{x\ e^{x\ y}}{x\ y}=0

    The problem is easily solved in polar coordinates setting...

    x= \rho\ \cos \theta

    y= \rho\ \sin \theta

    Kind regards

    \chi \sigma
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