Single Variable Calculus, Chapter 2, 2.4, Section 2.4, Problem 1

Given the graph of $f(x) = \frac{1}{x}$ to find a number $\delta$ such that if $|x - 2| < \delta $ then $\displaystyle |\frac{1}{x} -0.5 | < 0.2$






Referring to the graph, we will use the smaller value of $\delta$ to keep within a range of the $\varepsilon$. Thus


$
\begin{equation}
\begin{aligned}
\delta & \leq 2 - \frac{10}{7}\\
\delta & \leq \frac{14-10}{7}\\
\delta & \leq \frac{4}{7}

\end{aligned}
\end{equation}
$


This means that by keeping $x$ within $\displaystyle \frac{4}{7}$ of $2$, we are able to keep $f(x)$ within $0.2$ of $0.5$.

Although we chose $\displaystyle \delta = \frac{4}{7}$, any smaller positive value of $\delta$ would also have work.