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 $g$