Using polar co-ordinates to prove a bivariate limit exists at the origin.

In order to prove a limit at $\vtwo00$ exists for a function of form $𝑓 : 𝐷 \to ℝ$ , where $𝐷 \subseteq ℝ 2$ , we must show that the limit is the same from all directions of approach. In order to make this easier, we can convert the input from cartesian form $\vtwo𝑥𝑦$ to polar form $(𝑟, 𝜃)$ , where $𝑟 \in [0, \infty)$ and $𝜃 \in [0, 2 𝜋)$ . Since $(𝑥, 𝑦) \to (0, 0)$ corresponds to $𝑟 \to 0$ , it is only necessary to show the limit of the converted function as $𝑟 \to 0$ regardless of $𝜃$ . Often this will involve the Squeeze Theorem.

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