TITLE: Show that there is no limit
# !b2 Problem:
Show that the following limit does not exist.
> ;
                                             4  4
                                            x  y
                               lim        ----------
                          x, y -> (0, 0)    2    4 3
                                          (x  + y )

# !b3 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.

>