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Thread: Multivariate limit of a square root function

  1. #1
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    Multivariate limit of a square root function

    what is the limit of Sqrt[x^2 + y^2] as (x,y) approach (0,0)?

    I wrote 0 on the exam but I realized it probably doesn't exist because the sqrt fcn is defined only for positive values of x and y, what do you guys think?
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  2. #2
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    Re: Multivariate limit of a square root function

    I think $x^2 + y^2 \ge 0$ for all $x$ and $y$.

    Writing $x=r\cos\theta$ and $y=r\sin\theta$ should give you a good insight into the problem.
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  3. #3
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    Re: Multivariate limit of a square root function

    What you wrote on the exam is correct. If you look at the definition of a limit, it requires that "... for all $\displaystyle (x,y)$ in the domain of F with $\displaystyle 0<|(x,y)-(a,b)|<\delta$, $\displaystyle |F(x,y)-L|<\epsilon$."

    So you don't need an $\displaystyle (x,y)$ that makes $\displaystyle F(x,y)$ negative, you just need to look to see what F does to $\displaystyle (x,y)$ pairs near the origin.

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