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. |