Use the given graph of f to find a number δ such that if |x − 1| < δ then |f(x) − 1| < 0.2
Given : | x-1| <
then |f(x) − 1| < 0.2
To find the number \delta then is to find how close x has to be to 1.
In order for f(x) < 0.2 that is close to 1.
How close does x have to be to 1 (on either side), for f(x) to be between 0.8 and 1.2 ?
So, by the graph :
It is to clear that on the left side of x = 1, x can be within 0.3, but on the right side, it’d have to be within 0.1, of 1, for f(x) to be that close to 1.
f(x) is within 0.2 of 1 ( which is what | f(x) - 1 | < 0.2 is saying),
we take 0.1, which is sure to work on both sides.
Here used absolute values around each difference to show that the difference would work on both sides: a positive or negative difference would come out the same.
when we take the absolute values, Looks like it’s leading up to understanding “derivatives” or instantaneous slopes.