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Wolframalpha it says the limit does not exist

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data

    use definition for the limit of a function to show that
    [tex]\lim_{(x,y)\rightarrow (0,0)} xcos\frac{1}{y}[/tex]

    2. Relevant equations

    n/a

    3. The attempt at a solution

    fisrt, i assumed it the limit is 0(i dont know if it is true), but i showed it, but something bugging me, i put in wolframalpha it says the limit does not exist, http://www.wolframalpha.com/input/?i=lim+xcos(1/y),(x,y)->(0,0)

    so who is wrong here?
     
    Last edited: Aug 29, 2010
  2. jcsd
  3. Aug 29, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    Re: limit

    at a quicklook
    |cos(1/y)|<=1 for all y's,

    so you can choose x small enough to bring you close enough to zero. The function will oscillate with increasing frequency, but with decreasing magnitude as you approach the origin
     
  4. Aug 29, 2010 #3
    Re: limit

    so, it really approaches to 0, but how come wolfram-alpha say it does not exist. did i wrote anything wrong?
     
  5. Aug 29, 2010 #4
    Re: limit

    A limit of a function of several variables fails to exist if there are different limits from "different direction" i.e. from the "x" and "y" directions:

    Thus, you need to check if the limit as x -> 0 (treat y as a constant) of the function is the same as the limit as y -> 0 (treat x as a constant).

    Let's check limit from x direction:

    lim[x->0] x*cos(1/y) = 0*cos(const.) = 0

    But there is a problem with the limit of cos(1/y) when y->0.

    http://www.wolframalpha.com/input/?i=lim+cos(1/y),+y->0

    do you know why there is a problem with that y->0 limit? :)

    Good Luck.
     
  6. Aug 29, 2010 #5

    lanedance

    User Avatar
    Homework Helper

    Re: limit

    yeah its an interesting one...

    cos(1/y) is cleary undefined anywhere on the line y=0, so no limit on that line will exist for any function of cos(1/y)

    however, it does have the property that as you get close to x=0, |x||cos(1/y)|<=|x|which tends to zero

    also just to add, checking 2 startight line directions is suffcient to show a limit dne, but not to prove it exists
     
    Last edited: Aug 29, 2010
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