# Find the largest δ that works .

1. Jun 27, 2011

### -Dragoon-

Find the largest δ that "works".

1. The problem statement, all variables and given/known data
Find the largest δ that "works" for the given ϵ:
$\displaystyle \lim_{x\to1}2x = 2; ϵ = 0.1$

2. Relevant equations
N/A

3. 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.

2. Jun 27, 2011

### Dick

Re: Find the largest δ that "works".

I disagree with the textbook and I agree with you.

3. Jun 27, 2011

### -Dragoon-

Re: Find the largest δ that "works".

I see. Thanks.

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

4. Jun 27, 2011

### Dick

Re: Find the largest δ that "works".

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