Results 1 to 3 of 3

Math Help - Two variable limit question, prove with Paths

  1. #1
    Member CalcGeek31's Avatar
    Joined
    Aug 2008
    Posts
    83

    Two variable limit question, prove with Paths

    I don't know how to type with the math code, so this may look weird but bear with me please.


    x^3(y^2)
    lim _________
    (x,y) -> (0,0) x^2 + y^2



    I have already figured that the limit does not exist, because it comes out to 0/0, however, I have to prove it through paths. I have one path where I changed the y to another x, and got the answer of 0, however, i have been going through multiple paths and can not get anything but 0 to compare it with. Any help is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,969
    Thanks
    1009
    Quote Originally Posted by CalcGeek31 View Post
    I don't know how to type with the math code, so this may look weird but bear with me please.


    x^3(y^2)
    lim _________
    (x,y) -> (0,0) x^2 + y^2



    I have already figured that the limit does not exist, because it comes out to 0/0, however, I have to prove it through paths. I have one path where I changed the y to another x, and got the answer of 0, however, i have been going through multiple paths and can not get anything but 0 to compare it with. Any help is appreciated.
    Is it this?

    \lim_{(x, y) \to (0, 0)}\frac{x^3y^2}{x^2 + y^2}.


    I'd convert it to polars.

    Notice that x = r\cos{\theta}

    y = r\sin{\theta}

    and

    x^2 + y^2 = r^2.


    Also note that when (x, y) \to 0, then r \to 0.


    So we have

    \lim_{(x, y) \to (0, 0)}\frac{x^3y^2}{x^2 + y^2} = \lim_{r \to 0}\frac{r^3\cos^3{\theta}r^2\sin^2{\theta}}{r^2}

     = \lim_{r \to 0}r^3\cos^3{\theta}\sin^2{\theta}

     = 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member CalcGeek31's Avatar
    Joined
    Aug 2008
    Posts
    83
    I already know the limit does not exist, that is not the problem... its just the proving it doesn't, your logic makes sense to prove it exists, however, when starting the problem I was told it did not exist.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: November 7th 2011, 03:27 PM
  2. [SOLVED] 2-variable limit
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 6th 2011, 11:58 AM
  3. A two variable limit
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 7th 2011, 02:22 PM
  4. two variable limit
    Posted in the Pre-Calculus Forum
    Replies: 10
    Last Post: May 29th 2010, 06:10 AM
  5. quick question on paths
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 18th 2009, 01:59 PM

Search Tags


/mathhelpforum @mathhelpforum