The textbook I'm using only contains examples for very simple functions, like $$\lim_{(x,y) \rightarrow (0, 1)}{x^2 + y^2 + 2}$$
In this case, I can just break up the function into its composite parts:
$$\lim_{(x, y) \rightarrow (0, 1)}{x^2} + \lim_{(x, y) \rightarrow (0, 1)}{y^2} + \lim_{(x, y) \rightarrow (0, 1)}{2} = 0 + 1 + 2 = 3$$
For more complex functions, it has some examples for how to verify a given limit. But none for how to compute it in the first place. Suppose I am given something like one of the following problems:
$$\lim_{(x,y) \rightarrow (0,0)} \frac{e^{xy} - 1}{y}$$ $$\lim_{(x,y) \rightarrow (0,0)} \frac{\cos {(xy)} - 1}{x^2y^2}$$ $$\lim_{(x,y) \rightarrow (0,0)} \frac{xy}{x^2 + y^2 + 2}$$
How do I begin if I want to compute the limit? What is the general approach?