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 $\delta$.