# How do I find the limits of piecewise functions?

Jun 30, 2018

See below.

#### Explanation:

Suppose that the function is defined on the intervals

$\left({a}_{1} , {a}_{2}\right) \setminus \cup \left({a}_{2} , {a}_{3}\right) \setminus \cup \ldots \setminus \cup \left({a}_{n - 1} , {a}_{n}\right)$

where the interals can be open or closed, and ${a}_{1}$ and/or ${a}_{n}$ are possibly $\setminus \pm \setminus \infty$, and inside each interval $\left({a}_{i} , {a}_{i + 1}\right)$ $f \left(x\right)$ is defined as ${f}_{i} \left(x\right)$.

There are two cases: if you need to compute ${\lim}_{x \setminus \to c} f \left(x\right)$, where $c \setminus \in \left({a}_{i} , {a}_{i + 1}\right)$ for some $i$, then you simply compute ${\lim}_{x \setminus \to c} {f}_{i} \left(x\right)$, 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 ${f}_{i - 1} \left(x\right)$ in $\left({a}_{i - 1} , {a}_{i}\right)$ and ${f}_{i} \left(x\right)$ in $\left({a}_{i} , {a}_{i + 1}\right)$, you have

${\lim}_{x \setminus \to {a}_{i}} f \left(x\right) = {\lim}_{x \setminus \to {a}_{i}^{-}} {f}_{i - 1} \left(x\right) = {\lim}_{x \setminus \to {a}_{i}^{+}} {f}_{i} \left(x\right)$

So, the limit exists if and only if the left limit of ${f}_{i} \left(x\right)$ and the right limit of ${f}_{i - 1}$ exist and are the same.