Results 1 to 2 of 2

Thread: multivariable limits

  1. #1
    Banned
    Joined
    Oct 2008
    Posts
    71

    multivariable limits

    Prove that:

    .......$\displaystyle \lim_{ (x,y)\to (1,1)}{x^2y}=1$,by using the definition of the limit for a multivariable function
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by archidi View Post
    Prove that:

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

    therefore: $\displaystyle |x^2y-1|=|x^2(y-1)+x^2-1|<x^2|y-1|+(x+1)|x-1|<7\delta \leq \epsilon.$ hence $\displaystyle |x^2y-1| < \epsilon$ and we're done!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Do these 2 multivariable limits exist?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Oct 14th 2011, 05:09 AM
  2. Replies: 6
    Last Post: Mar 4th 2011, 08:27 AM
  3. evaluating limits - multivariable calculus
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Mar 8th 2010, 03:01 PM
  4. Multivariable Limits
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 12th 2010, 07:48 AM
  5. Multivariable Limits on TI-89
    Posted in the Calculators Forum
    Replies: 3
    Last Post: Oct 9th 2008, 03:48 AM

Search Tags


/mathhelpforum @mathhelpforum