Q:

Fidning multivariable limit

I'm having trouble in finding multi-variable limit and hope that an example like this could get me started on my work
$$\lim \limits_{(x, y) \to (2,0)} \frac{1-cosy}{xy^2}$$

A:

It is a limit in two variable but the indeterminate form that comes out is due to a one of them. Then we calculate:
$$\lim \limits_{(x, y) \to (2,0)} \frac{1-\cos y}{xy^2}=\frac{1}{2}\lim_{y\to 0}\frac{1-\cos y}{y^2}$$
But it is a well-known limit:
$$\lim_{t\to 0} \frac{1-\cos(t)}{t^2} = \frac{1}{2} $$
Then
$$\lim \limits_{(x, y) \to (2,0)} \frac{1-\cos y}{xy^2}=\frac{1}{2}\lim_{y\to 0}\frac{1-\cos y}{y^2}=\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$$

