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$} && \end{aligned} \end{equation}$