Results 1 to 2 of 2

Math Help - Functions of Several Variables

  1. #1
    Member
    Joined
    Apr 2008
    Posts
    191

    Functions of Several Variables

    Let F: R^2 \rightarrow R be defined by:



    Determine if the \lim_{(x,y) \to (0,0)} exists, if it does prove it.


    Here's my attempt:

    As (x,y) -> (0,0) along the y-axis, x=0:

    \lim_{(x,y) \to (0,0)} y^2 sin(\frac{1}{y^2})

    Along the x-axis:

    \lim_{(x,y) \to (0,0)} x^2 sin(\frac{1}{x^2})

    Along the line y=x:

    \lim_{(x,y) \to (0,0)} (x^2 +x) sin(\frac{1}{x^2 +x^2})

    I tried showing whether there is a common limit along different paths but I don't know how to finish this. Any help here is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    You could use the squeezing thm on the 3 limits you have to show all are 0.

    However showing you have a common limit along 3 different paths does not establish the limit. you would have to prove this for every possible path.

    You're method can only be used to show a limit does not exist--i.e. if the limit along different paths is different then the limit does not exist.

    However we from your results we suspect the limit is 0

    So let's return to the definition of limit. I'll use e for epsilon and d for delta

    We need to show

    whenever distance[(x,y) to 0] < d

    that is whenever x^2 + y^2 < d

    then |f(x,y) - L| < e


    |f(x,y) - L| = |f(x,y) - 0|

    =|(x^2 + y^2)sin1/[x^2+y^2] - 0 | < x^2 + y^2

    since |sin1/[x^2+y^2]|<1

    therefore if delta = e then

    |(x^2 + y^2)sin[1/(x^2+y^2)] | < e whenever x^2 + y^2 < d
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Functions of several variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 19th 2011, 01:03 AM
  2. Replies: 1
    Last Post: June 5th 2011, 04:57 PM
  3. Functions of Several variables Q
    Posted in the Calculus Forum
    Replies: 5
    Last Post: August 18th 2010, 07:01 AM
  4. Replies: 2
    Last Post: September 24th 2009, 10:43 PM
  5. Functions of several Variables
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 11th 2008, 08:44 PM

Search Tags


/mathhelpforum @mathhelpforum