Answer
See explanation
Work Step by Step
Step 1 First, we can try to directly recognize whether $f(x, y)$ is continuous at $(x_0, y_0)$ through: 1. Being a composition of continuous functions at $(x_0, y_0)$. 2. A sum, difference, or product of continuous functions at $(x_0, y_0)$. 3. Or a quotient of continuous functions, provided that the denominator is not zero at $(x_0, y_0)$. If $f(x, y)$ is continuous at $(x_0, y_0)$, then the limit exists, and \[ \lim_{{(x, y) \to (x_0, y_0)}} f(x, y) = f(x_0, y_0). \] Step 2 The second approach is to transform the function using polar or spherical coordinates. This can allow us to calculate and find the limit. Step 3 The third approach is to try to prove that $f(x, y)$ is not continuous by finding two different curves along which \[ \lim_{{(x, y) \to (x_0, y_0)}} f(x, y) \] has different values. In this situation, the limit doesn't exist.
