Results 1 to 7 of 7

Math Help - Investigating limits of functions of several variables

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    42

    Investigating limits of functions of several variables

    This is something that has bugged me for quite some time. I'm pretty comfortable working with limits of functions of several variables, but I always has this nagging feeling that the various methods - approaching along the axises, along some line y=kx, a parabola y=kx^2, finding some limiting function that's easier to evaluate at the point in question and so on - are all over the place. Like I'm just using experience and gut feeling to decide which to use. So basically I'd just like to throw it out there - do you have any rules of thumb, or perhaps even something more rigorous, to decide your approach to these types of problems?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2009
    Posts
    177
    Approching along the axises or other lines is a good way to prove divergence. If you want to prove convergence you often need to use the squeeze theorem.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Ted
    Ted is offline
    Member
    Joined
    Feb 2010
    From
    China
    Posts
    199
    Thanks
    1
    Polar Coordinates Technique is also a good way to prove convergence.
    Replace every x by  r cos(\theta) and every y by r sin(\theta)
    for the new limit, r will approach 0. You will deal with theta as a constant , that is , cos(theta), sin(theta) ... all are constants.

    To apply this technique, (x,y) must approach (0,0).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2010
    Posts
    42
    Yes, like I said I'm pretty comfortable with the various methods; it's the fact that I often can't motivate WHY I use a certain approach for a certain problem as my first approach that bugs me. To just take two very simple examples that I was asked about the other day (Don't know how to use the fancier way of writing):

    x/(x^2 + y^2), (x, y) approaches (0,0).
    Here I simply take the absolute values as it approaches the point along the axises. Whereas in

    (x^2*(y-1)^2) / (x^2 + (y-1))^2), (x, y) approaches (0, 1)
    I conclude that the function is less than or equal to x^2 and go from there.

    In both cases I of course end up with a useful answer, but I can't really explain WHY I took the approach I did. This list could be a mile long.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Ted
    Ted is offline
    Member
    Joined
    Feb 2010
    From
    China
    Posts
    199
    Thanks
    1
    By practice, you could select the best method to evaluate it.
    for your problem, along \displaystyle y=\sqrt{x} limit = 1 , along \displaystyle y=\frac{1}{x} limit = 0.
    Since we have different values for the limit along different paths, the limit does not exist.
    Practice Practice Practice !!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Oct 2010
    Posts
    42
    So basically, you agree with my view that there isn't really anything to it but, so to speak, getting to know a really wide range of problems and use that as a baseline? Don't really like that; I prefer it when you can explicitly say why you take a certain approach. Oh well, guess this is one of those times when math goes artsy.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Ted
    Ted is offline
    Member
    Joined
    Feb 2010
    From
    China
    Posts
    199
    Thanks
    1
    Exactly.
    The same thing for integrals as an example. You have many methods, and by practice you will know which method to use or at least to exclude some methods.

    You could Have "a certain approach" for finding derivatives. When you differentiate a function, you will simply determine its type and using the differentiation rule for this type.
    But you do not have this for integrals, evaluating limits .. etc
    Sometimes, determining the type itself could be hard. For example, Determining the type of an ordiniary differential equation sometimes is hard and even impossible.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. investigating an operator
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 16th 2012, 08:05 AM
  2. Replies: 1
    Last Post: June 5th 2011, 03:57 PM
  3. Limits of 2 Variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 9th 2009, 05:03 AM
  4. Limits of functions with 2 variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 29th 2008, 10:46 PM
  5. Investigating a Sequence
    Posted in the Calculus Forum
    Replies: 11
    Last Post: May 19th 2008, 05:51 PM

Search Tags


/mathhelpforum @mathhelpforum