Find the largest δ that works .

[-Dragoon-]

Find the largest δ that "works".

Homework Statement

Find the largest δ that "works" for the given ϵ:
\displaystyle \lim_{x\to1}2x = 2; ϵ = 0.1

Homework Equations

N/A

The Attempt at a Solution

Given ϵ > 0, then:
if 0 < |x - 1| < δ then |2x - 2| < ϵ
ϵ = 0.1, so, |2x - 2| < 0.1
Now to establish the connection:
|2x - 2| => |2 (x - 1)| => |2||x - 1| => 2|x - 1|
Therefore: 2|x - 1| < ϵ => |x - 1| < \frac{ϵ}{2} => |x - 1| <\frac{0.1}{2} =>|x - 1| < 0.05. The largest value that "works" for δ is 0.05 since if:
0 < |x - 1| < 0.05 then |2x - 2| < 0.1

But, in my textbook, the answer is 2ϵ = 0.2 as the largest value that "works" for δ. So, I just wanted to know what I did wrong in my calculations as the book only shows the solution and not the work. Thanks in advance.

 

[Dick]

I disagree with the textbook and I agree with you.

 

[-Dragoon-]

I disagree with the textbook and I agree with you.

I see. Thanks.

I never would have thought university textbooks ever made such errors, even though high school ones were rife with them.

 

[Dick]

I see. Thanks.

I never would have thought university textbooks ever made such errors, even though high school ones were rife with them.

The aren't that uncommon even at the university level. The same people are writing the textbooks.