Question: Of the following choices of δ, which is the largest thatcould be used successfully with arbitrary ε in anepsilon-delta proof of lim x -> 2 ( 1- 3x) = -5a) delta= 3 epsilonb) delta = epsilonc) delta = epsilon /2d) delta = epsilon/4e) delta = epsilon / 5
Of the following choices of δ, which is the largest thatcould be used successfully with arbitrary ε in anepsilon-delta proof of lim x -> 2 ( 1- 3x) = -5
a) delta= 3 epsilon
b) delta = epsilon
c) delta = epsilon /2
d) delta = epsilon/4
e) delta = epsilon / 5

We are tasked with finding the largest choice of \(\delta\) that can be used in the epsilon-delta proof for the limit:

\[
\lim_{x \to 2} (1 - 3x) = -5
\]

In an epsilon-delta proof, we need to find a \(\delta\) in terms of \(\epsilon\) such that:

\[
0 < |x - 2| < \delta \implies |(1 - 3x) - (-5)| < \epsilon
\]

### Step 1: Simplifying the expression
We start by simplifying the expression \(|(1 - 3x) - (-5)|\):

\[
|(1 - 3x) - (-5)| = |1 - 3x + 5| = |-3x + 6| = 3|x - 2|
\]

So, we want to make:

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

### Step 2: Solving for \(\delta\)
From the inequality \(3|x - 2| < \epsilon\), we divide both sides by 3:

\[
|x - 2| < \frac{\epsilon}{3}
\]

Thus, we can choose \(\delta = \frac{\epsilon}{3}\), which guarantees that \(0 < |x - 2| < \delta\) implies that \(|(1 - 3x) - (-5)| < \epsilon\).

### Step 3: Comparing choices of \(\delta\)
Now, we compare the given choices of \(\delta\) with \(\delta = \frac{\epsilon}{3}\):

- (a) \(\delta = 3\epsilon\)
- (b) \(\delta = \epsilon\)
- (c) \(\delta = \frac{\epsilon}{2}\)
- (d) \(\delta = \frac{\epsilon}{4}\)
- (e) \(\delta = \frac{\epsilon}{5}\)

Since the largest valid \(\delta\) based on our calculation is \(\frac{\epsilon}{3}\), none of the provided options directly matches \(\frac{\epsilon}{3}\). However, the largest \(\delta\) from the given choices that would work is:

\[
\delta = \epsilon / 2
\]

Thus, the correct answer is:

**(c) \(\delta = \epsilon / 2\)**

Would you like further details or have any questions?

Here are 5 related questions:
1. What is an epsilon-delta proof and why is it important in calculus?
2. How do you determine the appropriate \(\delta\) for a given function and limit?
3. Can you explain how to apply epsilon-delta proofs to polynomial functions?
4. What happens if you choose a \(\delta\) that is too large for the epsilon condition?
5. Why do we divide \(\epsilon\) by 3 in this particular limit problem?

**Tip**: Always try to manipulate the expression involving \(f(x)\) to match the \(\epsilon\) condition before solving for \(\delta\).

Of the following choices of δ, which is the largest that could be used successfully with arbitrary ε in an epsilon-delta proof of lim x -> 2 (1 - 3x) = -5?

a) delta = 3 epsilon
b) delta = epsilon
c) delta = epsilon / 2
d) delta = epsilon / 4
e) delta = epsilon / 5

Explore the epsilon-delta proof for the limit lim x -> 2 (1 - 3x) = -5. Learn how to choose the appropriate delta for any epsilon value and understand the steps behind epsilon-delta proofs in calculus. Suitable for undergraduate students.

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