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