Limits of functions with more than one variable

[mit_hacker]

Homework Statement

(Q) f(x,y) = 2xy/(x^2+y^2), (x,y) not = 0.
0 (x,y) = 0.
is continuous at every point except the origin.

(a) Substitute y=mx and then m=tan(theta) to show that f varies with the line's angle of inclination.

(b) Use the formula obtained in part (a) to show that the limit of f as (x,y) ---> (0,0) along the line y=mx varies from -1 to 1. depending on the angle of approach.

Homework Equations

The Attempt at a Solution

After doing all the substitutions, I get Sin(2theta). From that, it is clear that as the angle changes, so will the limit of f(x,y). Moreover, Sin(2theta) can only have values between -1 and 1. But, how does that mean that the limit of f as (x,y) ---> (0,0) along the line y=mx varies from -1 to 1. depending on the angle of approach.

Shouldn't it be "the limit of f as (x,y)---> (0,0) ALONG THE CURVE y=Sin(2theta)??

 

[Galileo]

You've already fixed the curve before you evaluated the limit.
You said y=mx, where m=tan(theta). So you're considering the value of f(x,y) for those points on that line and you find that the value doesn't depend on x and y at all, but has the constant value sin(2theta) along that line (minus the origin).
So clearly the limit along the line varies as you vary theta.

 

[mit_hacker]

Thanks!

Thanks a lot for that little tid-bit!