Limits of Multi-Variable Functions

Date: 11/12/2004 at 03:24:36
From: Tom
Subject: Limits and continuity

Find the limit, if it exists, or show that the limit doesn't exist.

  lim (x*y*cos(y))/(3*x^2 + y^2) as (x,y) ==> (0,0)

I don't really know how to approach this.  The main problem is that I
don't know what makes it so that the limit doesn't exist.  There was
something in our lecture notes where for these types of questions we
were meant to find the limit of the function using different paths.

For example, you would substitute y = 0 into the function and find the
limit as x approaches 0.  Doing that with this function I get 
lim 0/(3*x^2)?  I don't really know how to interpret this.

Date: 11/13/2004 at 18:09:12
From: Doctor Jordan
Subject: Re: Limits and continuity

Hi Tom,

For a limit to exist, the function must be approaching the same value 
no matter which way it is approaching the point we are taking the 
limit at.  For example, if we want to find the limit of a function 
z(x,y) as (x,y) approach the origin (0,0), then we could come at the 
origin from the y-axis, come at the origin from the x-axis, come at 
the origin from the line y = x, etc.  If we get different values for 
any of these approaches (i.e. the limit as you approach from, say the 
x-axis is different from the limit as you approach from the line y = 
x), then we say the limit does not exist at the point we're 
considering.

If we substitute y = 0 in, that means we are approaching along the x-
axis.  Does that make sense?  It's like we're saying "y is already 
there, x you come over now".

So you could try setting y = x (e.g. replace every y with an x and 
just look at the function as x approaches 0), or x = 0 and replace 
every x in the function with a 0 and just look at it as y approaches 
0.  You could try other approaches, but these are the easiest ones to 
check and the most common in exercises.  If you try those methods and 
still can't figure it out, write me back.  (Note: You should only 
start trying to prove that the limit doesn't exist when you've tried
to figure out a limit, but failed each time.)

- Doctor Jordan, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/16/2004 at 11:04:32
From: Tom
Subject: Thank you (Limits and continuity)

Thank you for your help. :)