Activity 1.7.2.
Consider a function that is piecewise-defined according to the formula
\begin{equation*}
f(x) = \begin{cases}3(x+2)+2 \amp \text{ for }-3 \lt x \lt -2 \\
\frac{2}{3}(x+2)+1 \amp \text{ for }-2 \le x \lt -1 \\
\frac{2}{3}(x+2)+1 \amp \text{ for }-1 \lt x \lt 1 \\
2 \amp \text{ for }x = 1 \\
4-x \amp \text{ for }x \gt 1
\end{cases}
\end{equation*}
Use the given formula to answer the following questions.
(a)
(b)
For each of the values \(a = -2, -1, 0, 1, 2\text{,}\) determine \(\displaystyle \lim_{x \to a^-} f(x)\) and \(\displaystyle \lim_{x \to a^+} f(x)\text{.}\)
(c)
For each of the values \(a = -2, -1, 0, 1, 2\text{,}\) determine \(\displaystyle \lim_{x \to a} f(x)\text{.}\) If the limit fails to exist, explain why by discussing the left- and right-hand limits at the relevant \(a\)-value.
(d)
For which values of \(a\) is the following statement true?
\begin{equation*}
\lim_{x \to a} f(x) \ne f(a)
\end{equation*}
(e)
On the axes provided, sketch an accurate, labeled graph of \(y = f(x)\text{.}\) Be sure to carefully use open circles (○) and filled circles (●) to represent key points on the graph, as dictated by the piecewise formula.
