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Math Help - multivariable limits

  1. #1
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    multivariable limits

    Prove that:

    ....... \lim_{ (x,y)\to (1,1)}{x^2y}=1,by using the definition of the limit for a multivariable function
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  2. #2
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    Quote Originally Posted by archidi View Post
    Prove that:

    ....... \lim_{ (x,y)\to (1,1)}{x^2y}=1,by using the definition of the limit for a multivariable function
    this is a good question! so suppose \epsilon > 0 is given. let \delta=\min \{1, \frac{\epsilon}{7} \}. now if \sqrt{(x-1)^2+(y-1)^2} < \delta, then: |x-1|< \delta, \ |y-1| < \delta. since \delta<1, we'll have 0<x<2 and thus x^2 < 4.

    therefore: |x^2y-1|=|x^2(y-1)+x^2-1|<x^2|y-1|+(x+1)|x-1|<7\delta \leq \epsilon. hence |x^2y-1| < \epsilon and we're done!
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