Question

Prove the lim(xy)-->(0,0) when f(x,y) = x/(x^2+y^2)

Answer 1

Prove *what*? that the limit does not exist?

Answer 2

@oldrin.bataku Show in detail what the limit really is

Answer 3

I think you are the perfect guy for that :)

Answer 4

There is no limit. Observe two different paths approaching \((0,0)\) along the horizontal axis, one from the left and one from the right where \(y=0\):$$\lim_{(x,0)\to(0,0)}\frac{x}{x^2+y^2}=\lim_{x\to 0^-}\frac{x}{x^2}=\lim_{x\to0^-}\frac1{x}=-\infty\\\lim_{(x,0)\to(0,0)}\frac{x}{x^2+y^2}=\lim_{x\to 0^+}\frac{x}{x^2}=\lim_{x\to0^+}\frac1{x}=+\infty$$

Answer 5

For the limit to exist, it must be the same *regardless* of the path taken. Here we observe two different paths to \((0,0)\) yielding *very* different limits!