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-Dragoon-

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**Find the largest δ that "works".**

## Homework Statement

Find the largest δ that "works" for the given ϵ:

[itex]\displaystyle \lim_{x\to1}2x = 2; ϵ = 0.1[/itex]

## Homework Equations

N/A

## The Attempt at a Solution

Given ϵ > 0, then:

if [itex]0 < |x - 1| < δ[/itex] then [itex]|2x - 2| < ϵ[/itex]

ϵ = 0.1, so, [itex]|2x - 2| < 0.1[/itex]

Now to establish the connection:

[itex] |2x - 2| => |2 (x - 1)| => |2||x - 1| => 2|x - 1|[/itex]

Therefore: [itex] 2|x - 1| < ϵ => |x - 1| < \frac{ϵ}{2} => |x - 1| <\frac{0.1}{2} =>|x - 1| < 0.05.[/itex] The largest value that "works" for δ is 0.05 since if:

[itex] 0 < |x - 1| < 0.05[/itex] then [itex]|2x - 2| < 0.1[/itex]

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.