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. |