Find the largest number δ such that if |x − 1| < δ, then |5x − 5| < ε, where ε = 0.1. δ ≤ repeat and determine δ with ε = 0.01.

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:

Implies that:

Well, let's start with the second expression:

We can take the 5 out of the absolute value:

Now we can divide both sides by 5:

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:

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:

link

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