- #1
Kitty Kat
- 13
- 0
Homework Statement
f(x,y) =
(xy) / (x2 + y4), when (x, y) ≠ (0,0)
0, when (x,y) = (0,0)
Homework Equations
Explicitly show that f(x,y) does not satisfy
lim h -> 0 [ E(v,h) / ||h|| ] = 0 when v = 0
(h, v, and 0 are all vectors; I'm not sure how to put a hat on them)
The Attempt at a Solution
I have no idea how to actually tackle this problem using multivariable calculus methods so I just compared the slopes instead (not sure if this is a valid approach either).
Further more I don't really get the concept of E(v,h). All my professor said is that it should approach 0 if the function is differentiable. I'd really appreciate an explanation of E(v,h) please (:
f(h) = f(0,0) + (df/dx(0,0) * x) + (df/dy(0,0) * y) + E(h)
f(h) = 0 + 0x + 0y + E(h)
f(h) = E(h)
Let f(x,y) = f(h,h)
f(h,h) = h / (1+h2)
f'(h) = (-h2 + 1) / (1+h2)2
||h|| = 2h2
f'(||h||) = 4h
lim h-> 0 [ f'(h) / f'(||h||) ] = ∞