Problem:Lef f be the function defined by |
> ;
2
x y
f(x, y) = -------
2 4
x + y
| when (x,y) is not (0,0) and define |
> ;
f(0, 0) = 0
|
Is f continuous at (0,0)? Explain.
|
Solution:
| If we use the path y=a x^m with x->0 we obtain, |
> Limit('x*a^2*x^(2*m)/(x^2+a^4*x^(4*m))',x=0) =
> limit(a^2*x^(2*m-1)/(1+a^4*x^(4*m-2)),x=0);
2 (2 m) 2 (2 m - 1)
x a x a x
lim -------------- = lim -----------------
x -> 0 2 4 (4 m) x -> 0 4 (4 m - 2)
x + a x 1 + a x
| when m=1/2 this last limit produces |
> ;
2
a
------
4
1 + a
| so the limit depends on a and thus it DNE. The function f is therefore not continuous at (0,0) since it doesn't have a limit at (0,0). |