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Math Help - limits in calculus in several variables

  1. #1
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    limits in calculus in several variables

    hey,

    can't seem to find this limit:

    lim((x,y)->(0,0)) ((1+x^2+y^2)^(1/(x^2+y^2+xy^2))

    the answer should be "e", but I really don't know how to get there.

    appreciate help thanks
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by mariannehp View Post
    hey,

    can't seem to find this limit:

    lim((x,y)->(0,0)) ((1+x^2+y^2)^(1/(x^2+y^2+xy^2))

    the answer should be "e", but I really don't know how to get there.

    appreciate help thanks
    use polar coordinates to get

    \displaystyle \lim_{r \to 0}[1+r^2]^{\frac{1}{r^2+r^3\cos(\theta)\sin^2(\theta)}}

    Now rewrite as

    \displaystyle \lim_{r \to 0}exp\left(\frac{1}{r^2+r^3\cos(\theta)\sin^2(\the  ta)} \ln\left(1+r^2 \right) \right)

    and use L'hospitials rule on the above.
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  3. #3
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    We also know that \displaystyle\lim _{h \to 0} \left( {1 + h} \right)^{\frac{1}{h}}  = e
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