# 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>