Question

use the given graph of f to find a number δ such that if |x - 1| < δ then |f(x) - 1| < 0.2.

Explanation:

Step1: Analyze left - hand side distance

We want to find the distance from \(x = 1\) to the \(x\) - value where \(f(x)=0.8\) and \(f(x) = 1.2\). Looking at the graph, when \(f(x)=0.8\), \(x = 1.1\), and when \(f(x)=1.2\), \(x=0.7\). The distance from \(x = 1\) to \(x = 1.1\) is \(|1.1 - 1|=0.1\), and the distance from \(x = 1\) to \(x = 0.7\) is \(|1 - 0.7| = 0.3\).

Step2: Determine the value of \(\delta\)

The definition of the limit requires that for \(|x - 1|\lt\delta\), \(|f(x)-1|\lt0.2\). We take the smaller of the two distances calculated above. So \(\delta=\min\{|1 - 0.7|,|1.1 - 1|\}\).

Answer:

\(0.1\)