Show that the following limit exists and find its value.
limit(x,y approaches 0,0) [ (y^3) / (|x| + y^2) ]
The Attempt at a Solution
this was on a test and this is what i did. i got 4/12 pts on this.
i did the limit along the x-axis (y=0)
f(x,0) = (0^3) / ( |x| + 0 ) = 0
so the limit = 0 along x axis
along y axis (x=0):
f(0,y) = (y^3) / (0 + y^2) = y
and the limit (y goes to 0) y = 0
along x = y^2:
f(y^2,y) = (y^3) / (y^2 + y^2) = (y^2*(y)) / (y^2(1+1)) and y^2 cancels leaving y/2
and the limit y goes to 0 of y/2 = 0 along x=y^2
therefore limit(x,y approaches 0,0) [ (y^3) / (|x| + y^2) ]