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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?

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    Sometimes, is common try to find some paths going to the origin with different limits. For example, $x=y\to 0$, $x=0,y\to 0$, etc... if you find two paths with different limits, the limit does not exists.2012-09-15
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    @Sigur: So if I try a few, and get the same limit, then I use the $\epsilon$-$\delta$ method to verify that it is indeed the limit?2012-09-15
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    Or you can try to modify your function to simplify it. See the answer below.2012-09-15

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