Multivariable Functions with No Limit

These are surfaces that I use when I teach multivariable calculus. They are functions of two variables that do not have a limit at the origin. Their equations are:

f₁(x, y) = ((x² – y²)/(x² + y²))²
f₂(x, y) = x²/(x² + y⁴)
f₃(x, y) = xy/(x² + y²)

Notice that if we define f₁(0, 0) = 1 and f₃(0, 0) = 0, then these two functions also have the property that both partial derivatives exist at the origin, but the function is not differentiable there.