# Wolframalpha it says the limit does not exist

1. Aug 29, 2010

### annoymage

1. The problem statement, all variables and given/known data

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

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. Aug 29, 2010

### lanedance

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

3. Aug 29, 2010

### annoymage

Re: limit

so, it really approaches to 0, but how come wolfram-alpha say it does not exist. did i wrote anything wrong?

4. Aug 29, 2010

### gomunkul51

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.

5. Aug 29, 2010

### lanedance

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