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.