Limits of functions of two variables

[legadema]

first I'm a new user and i want to say thank u for such a forum
my question is for two trayectories
1st y = x
2nd x = 0
meant to calculate the limit of (3x^2*y)/(x^2 + y^2) as (x,y) approaches (0,0)

first trayectory:
lim (3x^2*y)/(x^2 + y^2) = 3x^3/2x^2 = (3/2)x

second trayectory:
lim (3x^2*y)/(x^2 + y^2) = 0

and my teacher says that since the two limits are different the limit doesn't exists.
but is this true? i think they are not different, since (3/2)x = 0 as x approaches 0
please clarify this to me

 

[lanedance]

1st traj
(x,y) -> (x,x)
i guess you mean
y = x
(3x^2*y)/(x^2 + y^2) = (3x^3)/(2x^2) = (3/2)x

2nd traj doesn't quite make sense
(x,y) -> (0,x)?
however trajectories along y=0 or x =0 both give a limit of zero

 

[legadema]

yes man i meant on 1st traj y=x and on 2nd x=0