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Math Help - Two variables calculus-not too hard

  1. #1
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    Two variables calculus-not too hard

    1. Given the function:
     f(x,y) = \frac{xy^{2}}{x^{4}+y^{2}}
    Does the limit  \lim_{(x,y)\to(0,0)} f(x) exist?


    2. For positive  \alpha , \beta , show that:
     \lim_{(x,y)\to(0,0)} x^{\alpha}y^{\beta}ln(x^{4}+y^{2}) =0


    I'll be delighted to get some guidance in these two...
    In the first one I've tried to devide the function by y^{2}, but I couldn't understand how to continue...
    About the second one- I think I should do something wil l'hospital's law or somehing... But I can't figure out how...

    Thanks!
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    1. Given the function:
     f(x,y) = \frac{xy^{2}}{x^{4}+y^{2}}
    Does the limit  \lim_{(x,y)\to(0,0)} f(x) exist?


    2. For positive  \alpha , \beta , show that:
     \lim_{(x,y)\to(0,0)} x^{\alpha}y^{\beta}ln(x^{4}+y^{2}) =0


    I'll be delighted to get some guidance in these two...
    In the first one I've tried to devide the function by y^{2}, but I couldn't understand how to continue...
    About the second one- I think I should do something wil l'hospital's law or somehing... But I can't figure out how...

    Thanks!
    For the first, use the sandwich theorem:

    \left| \frac{xy^2}{x^4+y^2} \right| \leq \frac{xy^2}{y^2}=x.
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