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.