# Find the largest δ that works .

• -Dragoon-
In summary, the largest value of δ that "works" for the given ϵ = 0.1 is 0.05. There may be a discrepancy with the answer given in the textbook, which is 0.2, but this could be due to an error in the textbook.
-Dragoon-
Find the largest δ that "works".

## Homework Statement

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

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.

I disagree with the textbook and I agree with you.

Dick said:
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.

Retribution said:
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.

## What does "Find the largest δ that works" mean?

"Find the largest δ that works" refers to finding the largest difference or change in a variable that still yields a successful or valid result in a scientific experiment or calculation.

## Why is it important to find the largest δ that works?

Finding the largest δ that works is important because it helps to determine the limits or boundaries of a system or experiment. It allows scientists to understand the range of values for which their findings or conclusions are applicable.

## How do scientists determine the largest δ that works?

Scientists determine the largest δ that works through a process of trial and error, adjusting the variable or condition in question until the desired result is achieved. This may involve running multiple experiments or simulations and analyzing the data to find the optimal value for δ.

## What factors can affect the largest δ that works?

The largest δ that works can be affected by a variety of factors, including the specific conditions or variables being studied, the accuracy and precision of the measurements or instruments used, and the limitations of the experimental setup or model.

## Are there any limitations to using the largest δ that works in scientific research?

Yes, there are limitations to using the largest δ that works in scientific research. It is important for scientists to carefully consider the potential impact of outliers or extreme values on their findings, as well as the potential for bias or error in their methods. Additionally, the largest δ that works may not necessarily be the most accurate or representative value for a given phenomenon, and further research may be needed to fully understand its implications.

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