Another service from Omega

Show that there is no limit


*****

Problem:

Show that the following limit does not exist.

> ;

                                             4  4
                                            x  y
                               lim        ----------
                          x, y -> (0, 0)    2    4 3
                                          (x  + y )


Solution:


We only need to find two paths ariving to (0,0) along which the limits are different. We consider the paths y=x and x=y^2. When y=x with x->0 we have,

> Limit(`x^4*x^4`/`(x^2+x^4)^3`,x=0) = Limit(x^4*x^4/(x^2+x^4)^3,x=0);

                                                      8
                             x^4*x^4                 x
                    lim    ----------- =  lim    ----------  = 0
                   x -> 0  (x^2+x^4)^3   x -> 0    2    4 3
                                                 (x  + x )

but along x=y^2 with y->0 we have,

> Limit(`y^8*y^4`/`(y^4+y^4)^3`,y=0) = Limit(y^8*y^4/(y^4+y^4)^3,y=0);

                                 y^8*y^4
                        lim    ----------- =  lim    1/8 = 1/8
                       y -> 0  (y^4+y^4)^3   y -> 0

Since the two paths produce different limits then the limit does not exist.


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Fri Mar 23 14:40:33 EST 2001