Let be defined as follows:
for all (x,y) unless
for all (x,y) where
Show that as on any straight line through (0,0)
Determine if exist as
just need some explaination on this question
First check that if you approach (0, 0) along x = 0 or y = 0, then x^4 < y < x^2 is never the case, so the limit is 0.
If the line through (0, 0) is not vertical or horizontal, then it is described by an equation y = kx for some k <> 0. Therefore, f(x, y) = 1 on this line iff x^4 < kx < x^2.
There are several cases to consider here. Assume that x > 0. Then we can divide by x without changing the inequality: x^3 < k < x. So, iff . If k >= 1, then this is never the case, so approaching (0, 0) along such line from the right has limit 0. If k < 1, then such limit is also 0 because eventually x <= k. However, as k becomes smaller and smaller, you have f(x, y(x)) = 1 for x that are arbitrarily close to 0. This means that in every neighborhood around (0, 0) there are points (x, y) such that f(x,y) = 0 and other points (x', y') such that f(x',y') = 1.
There may be some cases left to consider.