Question

Preliminary analysis of the problem (guessing a value for 𝛿). Let 𝜀 be a given positive number. We want to find a number 𝛿 such that
if 0 < |x − 6| < 𝛿 then |(2x − 5) − 7| < 𝜀.
But
|(2x − 5) − 7| = |2x − 12| = 2


Correct: Your answer is correct.


.
Therefore, we want 𝛿 such that
if 0 < |x − 6| < 𝛿 then 2


Incorrect: Your answer is incorrect.
Your answer cannot be understood or graded. More Information

< 𝜀
that is, if 0 < |x − 6| < 𝛿 then


Incorrect: Your answer is incorrect.


<
𝜀
2
.

This suggests that we should choose 𝛿 = 𝜀/2.

Answer 1

It looks like you're trying to analyze the problem of finding a suitable \(\delta\) in order to satisfy the condition for a limit, specifically for the function \(f(x) = 2x - 5\) as \(x\) approaches \(6\). The goal is to ensure that \(|f(x) - 7| < \epsilon\) whenever \(|x - 6| < \delta\).

Let's break it down step-by-step:

1. **Determine the expression**:

We want to analyze \(|f(x) - 7|\):

\[
f(x) = 2x - 5
\]
Therefore,

\[
|f(x) - 7| = |(2x - 5) - 7| = |2x - 12| = 2|x - 6|
\]

2. **Set the goal**:

We want this expression to be less than \(\epsilon\):

\[
|2x - 12| < \epsilon
\]

From our earlier calculation:

\[
2|x - 6| < \epsilon
\]

3. **Manipulate the inequality**:

To make this easier, we can divide both sides by \(2\):

\[
|x - 6| < \frac{\epsilon}{2}
\]

4. **Choose \(\delta\)**:

Thus, we can choose:

\[
\delta = \frac{\epsilon}{2}
\]

5. **Conclusion**:

In summary, if we choose \(\delta = \frac{\epsilon}{2}\), then whenever \(0 < |x - 6| < \delta\), it will guarantee that:

\[
|(2x - 5) - 7| < \epsilon
\]

This procedure is common in proving limits. The approach you've taken is a good way to analyze the continuity of the function at the given point. If you have additional questions or need clarification on this topic, feel free to ask!