Suppose that the function is defined on the intervals
(a_1,a_2) \cup (a_2,a_3) \cup ... \cup (a_{n-1},a_n)
where the interals can be open or closed, and a_1 and/or a_n are possibly \pm\infty, and inside each interval (a_i, a_{i+1}) f(x) is defined as f_i(x).
There are two cases: if you need to compute lim_{x \to c} f(x), where c \in (a_i, a_{i+1}) for some i, then you simply compute lim_{x \to c} f_i(x), as if it was a "normal" function.
The only special case is when you want to compute the limit with x tending towards a border point. In this case, you simply compute the left and right limits, using the correct definitions: if you have f_{i-1}(x) in (a_{i-1},a_i) and f_i(x) in (a_i,a_{i+1}), you have
lim_{x\to a_i} f(x) = lim_{x\to a_i^-}f_{i-1}(x) = lim_{x\to a_i^+}f_i(x)
So, the limit exists if and only if the left limit of f_i(x) and the right limit of f_{i-1} exist and are the same.
