Single Variable Calculus, Chapter 2, Review Exercises, Section Review Exercises, Problem 10

Determine $\displaystyle \lim \limits_{v \to 4^+} \frac{4 - v}{| 4 - v|}$

Recall


$
\begin{equation}
\begin{aligned}

|x| =& \left\{ \begin{array}{ccc}
x & \text{ if } & x \geq 0 \\
-x & \text{ if } & x < 0
\end{array} \right.
\end{aligned}
\end{equation}
$


So,


$
\begin{equation}
\begin{aligned}
|4 - v| =& \left\{ \begin{array}{ccc}
-(4-v) & \text{ if } & v \geq 4 \\
(4-v) & \text{ if } & v < 4
\end{array} \right.

\end{aligned}
\end{equation}
$

Since $v$ approaches 4 from the right-hand limit. We use $-(4 - v)$ to eliminate the absolute sign.


$
\begin{equation}
\begin{aligned}

\lim \limits_{v \to 4^+} \frac{4 - v}{| 4 - v|} &= \lim \limits_{v \to 4^+} \frac{\cancel{4 - v}}{-\cancel{(4 - v)}} = \frac{1}{-1}
&& \text{Cancel out like terms}
\\
\\
& \fbox{$= -1$}
&&


\end{aligned}
\end{equation}
$