Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 45

Suppose that the signum function, denoted by sgn is defined by:

$\text{sgn }x = \left\{ \begin{array}{c} -1 & \text{if} & x < 0\\ 0 & \text{if} & x = 0\\ 1 & \text{if} & x > 0 \end{array} \right.$


(a) Sketch the graph of the function sgn $x$
(b) Determine each of the following limits and explain if the limit does not exist


$\begin{equation} \begin{aligned} (i) & \lim\limits_{x \to 0^+} \text{ sgn }x & (ii) & \lim\limits_{x \to 0^-} \text{ sgn }x\\ (iii) & \lim\limits_{x \to 0} \text{ sgn }x & (iv) & \lim\limits_{x \to 0} |\text{ sgn } x| \end{aligned} \end{equation}$



a.)



b.)

$(i)$ Referring to the graph given, the $\lim\limits_{x \to 0^+} \text{sgn} x = 1$
$(ii)$ Referring to the graph given, the $\lim\limits_{x \to 0^-} \text{sgn} x = -1$
$(iii)$ Referring to the graph given, the $\lim\limits_{x \to 0} \text{sgn} x$, does not exist because the left and right hand limits are different.
$(iv)$ Referring to the graph given, the $\lim\limits_{x \to 0} |\text{sgn} x | = 1$ for all values of $x$ except 0. Therefore, the limit is 1.