Another service from Omega

Show that limit DNE


*****

Problem:

Show that the following limit DNE (Does Not Exist):

> ;

                                             2
                                            x  + y
                                 lim        -------
                            x, y -> (0, 0)   2    2
                                            x  + y


Solution:


Consider paths of the form y=a x^m with m>0 as x->0.

> Limit('(x^2+a*x^m)/(x^2+a^2*x^(2*m))',x=0) =
> limit((1+a*x^(m-2))/(1+a^2*x^(2*m-2)),x=0);

                         2      m                      (m - 2)
                        x  + a x                1 + a x
               lim    -------------- =  lim    -----------------
              x -> 0   2    2  (2 m)   x -> 0       2  (2 m - 2)
                      x  + a  x                1 + a  x

and for m=2 we get this limit to be 1+a. This depends on a so the limit DNE.


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Fri Mar 23 15:09:57 EST 2001