Results 1 to 6 of 6

Math Help - Systematic approach for multi variable limits?

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    40

    Systematic approach for multi variable limits?

    I feel like my knowledge in finding limits of multi variable functions has a few loose ends and was wondering if anyone here might help me tie them up. I have no problem performing the various tests; if for example we wish to evaluate a limit at the origin of a function f(x,y) you can let the f approach (0,0) along the axises, a line y=kx, a curve y=x^2 and so on. But that's kind of the thing. I feel like I'm never entirely sure what tests I should use or when I have enough of them agree on a specific limit to be able to conclude that that is indeed the limit. I generally resort to graphing and that works, of course, but is there a systematic way of approaching it with just the function definition, a piece of paper, a pen and a glaring lack of artistic talent?
    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: Systematic approach for multi variable limits?

    Quote Originally Posted by Scurmicurv View Post
    I feel like my knowledge in finding limits of multi variable functions has a few loose ends and was wondering if anyone here might help me tie them up. I have no problem performing the various tests; if for example we wish to evaluate a limit at the origin of a function f(x,y) you can let the f approach (0,0) along the axises, a line y=kx, a curve y=x^2 and so on. But that's kind of the thing. I feel like I'm never entirely sure what tests I should use or when I have enough of them agree on a specific limit to be able to conclude that that is indeed the limit. I generally resort to graphing and that works, of course, but is there a systematic way of approaching it with just the function definition, a piece of paper, a pen and a glaring lack of artistic talent?
    An approach working about 'cent per cent' is the following: use polar coordinates setting...

    x= r\ \cos \theta

    y= r\ \sin \theta (1)

    ... and then find the limit for r to 0 independently from \theta. If the limit exists then You are ok!...



    Marry Christmas from Serbia

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

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,404
    Thanks
    1293

    Re: Systematic approach for multi variable limits?

    Quote Originally Posted by chisigma View Post
    An approach working about 'cent per cent' is the following: use polar coordinates setting...

    x= r\ \cos \theta

    y= r\ \sin \theta (1)

    ... and then find the limit for r to 0 independently from \theta. If the limit exists then You are ok!...



    Marry Christmas from Serbia

    \chi \sigma
    If the limit exists and is a number, then you have the limit.

    If the limit exists and depends on \displaystyle \begin{align*} \theta \end{align*}, then you know that the limit is going to change depending on which path you take, which means the limit does not exist.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2010
    Posts
    40

    Re: Systematic approach for multi variable limits?

    Well, that makes sense. Just as a follow-up then, there is no equivalently sure-fire way of doing it in Cartesian coordinates? How come?
    Follow Math Help Forum on Facebook and Google+

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

    Re: Systematic approach for multi variable limits?

    Quote Originally Posted by Scurmicurv View Post
    Well, that makes sense. Just as a follow-up then, there is no equivalently sure-fire way of doing it in Cartesian coordinates? How come?
    Sometimes something similar to the approach in the following example works. You have the function...

    f(x,y)=\begin{cases}\frac{x y^{2}}{x^{2}+y^{2}} &\text{if}\ (x,y) \ne (0,0)\\ 0 &\text{if}\ (x,y) = (0,0)\end{cases} (1)

    ... and You should find \lim_{(x,y) \rightarrow (0,0)} f(x,y). In that case You consider that |x y^{2}| \le |x|\ |x^{2}+y^{2}| and You can write...

    |\frac{x y^{2}}{x^{2}+y^{2}}| \le |x| \implies \lim_{(x,y) \rightarrow (0,0)} f(x,y) \le \lim_{(x,y) \rightarrow (0,0)} |x|= 0 (2)

    The use of polar coordinates however is also in that case easier and faster and doesn't require to search more or less complicated 'excamotages'. The pratical limit of that approach is when the function has more than two variables. For a three variables function may be that the use of r, \theta and \phi sferical coordinates is possible. But if the variables are four or more?...




    Marry Christmas from Serbia

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

  6. #6
    Junior Member
    Joined
    Oct 2010
    Posts
    40

    Re: Systematic approach for multi variable limits?

    Hm. Well, food for thought, if nothing else, and those loose ends I mentioned at the outset don't feel quite so random now. Thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Multi-variable pdf.
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: December 3rd 2011, 12:31 PM
  2. Multi-variable limit
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 19th 2011, 08:35 AM
  3. Multi-variable Culculus limits (prove)
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 27th 2010, 01:05 PM
  4. how to approach induction with more then one variable?
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: April 14th 2010, 02:47 PM
  5. Multi variable integration?
    Posted in the Calculus Forum
    Replies: 7
    Last Post: December 30th 2009, 08:20 PM

Search Tags


/mathhelpforum @mathhelpforum