Single Variable Calculus, Chapter 2, 2.4, Section 2.4, Problem 5

We need to use a graph to find a number $\delta$ such that if $\displaystyle |x - \frac{\pi}{4} | < \delta $ then $|\tan x - 1| < 0.2$









First, we will get the values of $x$ that intersect at the given curve to their corresponding $y$ values. Let $x_L$ and $x_R$
are the values of $x$ from the left and right of $\displaystyle \frac{\pi}{4}$ respectively.

$
\begin{equation}
\begin{aligned}
y & = \tan x_L &
y & = \tan x_R\\

0.8 & = \tan x_L &
1.2 & = \tan x_R\\

x_L & = \tan^{-1}[0.8] &
x_R & = \tan^{-1}[1.2]\\

x_L & = 0.6741 &
x_R & = 0.8761
\end{aligned}
\end{equation}
$


Now, we can determine the value of $\delta$ by checking the values of $x$ that would give a smaller distance to $\displaystyle \frac{\pi}{4}$.


$
\begin{equation}
\begin{aligned}
\frac{\pi}{4} - x_L = \frac{\pi}{4} - 0.6747 & = 0.1107\\
\frac{\pi}{4} - x_R =0.8761 - \frac{\pi}{4} & = 0.0907
\end{aligned}
\end{equation}
$


Hence,
$\quad \delta \leq 0.0907$

This means that by keeping $x$ within $0.0907$ of $\displaystyle \frac{\pi}{4}$, we are able to keep $f(x)$ within $0.2$ of $1$.

Although we chose $\delta = 0.0907$, any smaller positive value of $\delta$ would also have work.