Hi, here we want to "finish" a delta-epsilon proof.
We will find that the maximum delta is: δ = 0.02
We just want to find the value of δ such that:
[tex]|x - 1| < \delta[/tex]
Implies that:
[tex]|5x - 5| < \epsilon[/tex]
Well, let's start with the second expression:
[tex]|5x - 5| < \epsilon[/tex]
We can take the 5 out of the absolute value:
[tex]5*|x - 1| < \epsilon[/tex]
Now we can divide both sides by 5:
[tex]|x - 1| < \epsilon/5[/tex]
Now, if you look at the above expression, in the left side we have the same thing as in the delta inequality.
So for now we have:
[tex]|x - 1| < \epsilon/5[/tex]
[tex]|x - 1| < \delta[/tex]
From this is easy to conclude that the maximum value that delta can take is:
δ = ε/5
And we know that ε = 0.1, then we have:
δ = 0.1/5 = 0.02
δ = 0.02
The maximum value that delta can take is 0.02
Then we need to repeat it for ε = 0.01, so we get:
δ = 0.01/5 = 0.002
δ = 0.002
If you want to learn more, you can read:
https://brainly.com/question/14795650
