Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 48

Given the function
$
g(x) = \left\{
\begin{array} {c}
x & \text{ if } x < 1\\
3 & \text{ if } x = 1\\
2 - x^2 & \text{ if } 1 < x \leq 2\\
x -3 & \text{ if } x > 2
\end{array}
\right.
$

a.) Find each of the following limits if it exists

$
\begin{equation}
\begin{aligned}
(i) & \lim\limits_{x \to 1^-} g(x) &
(ii) & \lim\limits_{x \to 1} g(x) &
(iii) & g(1)\\

(iv) & \lim\limits_{x \to 2^-} g(x) &
(v) & \lim\limits_{x \to 2^+} g(x) &
(vi) & \lim\limits_{x \to 2} g(x)
\end{aligned}
\end{equation}
$


b.) Sketch the graph of $g$


Answers:
a.)


$
\begin{equation}
\begin{aligned}
& (i) \lim\limits_{x \to 1^-} g(x) && = \lim\limits_{x \to 1^-} x \quad = 1\\
& (ii) \lim\limits_{x \to 1} g(x) && \text{We evaluate first the right limit of the function to see whether the } \lim\limits_{x \to 1} g(x) \text{ exist}\\
& \phantom{x} && \lim\limits_{x \to 1^+} g(x) \quad = \lim\limits_{x \to 1^+} (2-x^2) \quad = 2-(1)^2 \quad = 1\\
& \phantom{x} && \text{The left and right hand limits are equal. Therefore, } \lim\limits_{x \to 1} g(x) \text{ exist and is equal to 1}\\
& \phantom{x} && \lim\limits_{x \to 1} g(x) = 1\\
& (iii) g(1) & & g(1) = 3\\
& (iv) \lim\limits_{x \to 2^-} g(x)&& = \lim\limits_{x \to 2} (2-x^2) \quad= 2 - (2)^2 \quad= -2\\
& (v) \lim\limits_{x \to 2^+} g(x) && = \lim\limits_{x \to 2} (x-3)\quad = 2 - 3 \quad = -1\\
& (vi) \lim\limits_{x \to 2} g(x) && \text{Does not exist because the left and right hand limits are different}
\end{aligned}
\end{equation}
$


b.) Sketch the graph of $g$