Results 1 to 3 of 3

Math Help - Limits and Continuity - two variable function

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    133

    Limits and Continuity - two variable function

    Hello MHF,
    Needing some help with this question,

    What condition must the nonnegative integers m,n,p satisfy to guarantee that \lim_{(x,y)\to(0,0)} \frac{x^my^n}{(x^2+y^2)^p} exists? Prove your answer. [You may assume that m,n,p are not all zero.]
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    With the substitutions...

    x= \rho\cdot \cos \theta

    y= \rho\cdot \sin \theta (1)

    ... the limit becomes...

    \lim_{(x,y) \rightarrow (0,0)} \frac{x^{m}\cdot y^{n}}{(x^{2} + y^{2})^{p}} = \lim_{\rho \rightarrow 0} \frac{\rho^{m+n}}{\rho^{2p}} (\cos^{m} \theta \cdot \sin^{n} \theta) (2)

    The limit (2) exist if and only if it is independent from \theta and this is true only if...

    m+n>2\cdot p (3)

    In this case the limit is 0...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,453
    Thanks
    1868
    Blast you! You got in before I could say it!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Two-Variable Limit and Continuity
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 7th 2010, 04:02 AM
  2. Replies: 3
    Last Post: October 24th 2009, 03:54 AM
  3. Continuity of a 2 variable function
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 1st 2009, 01:05 PM
  4. limits and continuity
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: February 20th 2009, 02:46 PM
  5. Limits and Continuity
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 2nd 2008, 12:25 AM

Search Tags


/mathhelpforum @mathhelpforum