二元函数的极限

A two-variable limit must survive every approach

One bad path vetoes the whole limit

Polar coordinates can turn a veto test into a proof

Limits follow the same arithmetic as before

What is $\lim_{(x,y) \to (0,0)} \dfrac{x^2 - y^2}{x^2 + y^2}$?

A student tests $f(x, y) = \dfrac{xy}{x^2 + y^2}$ near $(0, 0)$ along the two axes, finds the value $0$ both times, and concludes the limit is $0$. What is the correct assessment?

Using polar coordinates, what bound controls $\left|\dfrac{x^3 - y^3}{x^2 + y^2}\right|$ near the origin, and what limit follows?

Compute $\lim_{(x,y) \to (2,3)} \dfrac{x^2 y - 1}{x + y}$ and give the result as a fraction like 11/5.

Compute $\lim_{(x,y) \to (0,0)} \dfrac{x^2 + 2y^2}{x^2 + y^2}$ along the parabola $y = x^2$ and give the value it produces, as a fraction like 3/2 or a whole number.