Results 1 to 5 of 5

Math Help - multivariable limit.

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    146

    multivariable limit.

    I just have a general question on evaluating multivariable limits approaching the origin (0,0). If i determine that approaching from the x-axis and y-axis yields the same limit, my instructor said that it doesn't have to necessarily exist as it may not exist from another path.

    So my question is, how do I go about finding such paths since there are literally an infinite number of them that approach the origin.
    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

    Re: multivariable limit.

    Quote Originally Posted by Kuma View Post
    I just have a general question on evaluating multivariable limits approaching the origin (0,0). If i determine that approaching from the x-axis and y-axis yields the same limit, my instructor said that it doesn't have to necessarily exist as it may not exist from another path.

    So my question is, how do I go about finding such paths since there are literally an infinite number of them that approach the origin.
    An efficient approach is to convert f(x,y) in polar form f(\rho, \theta) with the substitution...

    x= \rho\ \cos \theta

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

    ... and then find the limit if \rho tends to 0 of f(\rho, \theta) taking care that the limit must be independent from \theta ...

    Kind regards

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

  3. #3
    Member
    Joined
    Nov 2008
    Posts
    146

    Re: multivariable limit.

    Hi. Thanks for the reply, I haven't learned polar forms yet. Is there any other way to approach this to make sure the limit does indeed exist?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2010
    From
    Clarksville, ARk
    Posts
    398

    Re: multivariable limit.

    Quote Originally Posted by Kuma View Post
    Hi. Thanks for the reply, I haven't learned polar forms yet. Is there any other way to approach this to make sure the limit does indeed exist?
    You're doing multivariable limits and you don't know polar coordinates? Are you sure?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,328
    Thanks
    702

    Re: multivariable limit.

    you can do it directly from the definition of the limit, but this is often "messy". but this amounts to the same thing as the polar coordinates version, really.

    the idea is: that the magnitude of |f(x,y) - L| has to go to 0 as we consider the value of |f(x,y) - L| on "smaller and smaller" circles (well, disks, actually) surrounding the origin.

    a common example is the following function:

    f(x,y) = \frac{x^2-y^2}{x^2 + y^2}. on a circle of radius r, f(x,y) = (1/r^2)(x^2 - y^2).

    it is not hard to show that on this circle, f takes on the values 1 and -1, (for example at (r,0) and (0,r)) so as we "shrink the circles", the value of f(x,y) does not converge.

    contrast this with the behavior of g(x,y) = x^2 + y^2. on a circle of radius r, g(x,y) = r^2, and as we shrink the circles (approaching (0,0) from "all directions at once") r^2 \to 0.

    formally, let ε > 0, and choose δ = √ε.

    then |(x,y) - (0,0)| < δ --> |(x,y)| < δ, that is:

    \sqrt{x^2 + y^2} < \delta \implies x^2 + y^2 = g(x,y) < \delta^2 = \epsilon, so

    |g(x,y) - 0| = |g(x,y)| = g(x,y) < \epsilon (since g is always non-negative).

    given an epsilon, we can find a delta, so

    \lim_{(x,y) \to (0,0)} g(x,y) = 0
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Multivariable limit.
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 18th 2011, 04:36 AM
  2. multivariable limit
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 26th 2011, 06:13 PM
  3. Multivariable Limit Problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 23rd 2010, 07:08 PM
  4. Limit of multivariable function
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 16th 2009, 07:28 PM
  5. multivariable limit
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 19th 2008, 01:51 PM

Search Tags


/mathhelpforum @mathhelpforum