The problem you uploaded asks to use the formal definition of a limit to find the largest values of $\delta$ that correspond to the given $\epsilon$ values for the limit

$\lim_{{x \to 2}} (4x - 4) = 4.$

Step-by-Step Solution:

We know from the definition of a limit that for any $\epsilon > 0$, there exists a $\delta > 0$ such that:

$0 < |x - 2| < \delta \implies |(4x - 4) - 4| < \epsilon.$

Step 1: Simplifying the Expression

Start by simplifying $|(4x - 4) - 4|$:

$|(4x - 4) - 4| = |4x - 8| = 4|x - 2|.$

So, the inequality becomes:

$4|x - 2| < \epsilon.$

Step 2: Solving for $\delta$

We want to solve for $\delta$. Since $4|x - 2| < \epsilon$, divide both sides of the inequality by 4:

$|x - 2| < \frac{\epsilon}{4}.$

Therefore, we can choose $\delta = \frac{\epsilon}{4}$. This means that for each $\epsilon$, the corresponding $\delta$ is $\frac{\epsilon}{4}$.

Step 3: Calculating $\delta$ for each $\epsilon$

• For $\epsilon = 0.5$: $\delta = \frac{0.5}{4} = 0.125.$

• For $\epsilon = 0.1$: $\delta = \frac{0.1}{4} = 0.025.$

• For $\epsilon = 0.05$: $\delta = \frac{0.05}{4} = 0.0125.$

Final Results:

• For $\epsilon = 0.5$, $\delta \leq 0.125$.
• For $\epsilon = 0.1$, $\delta \leq 0.025$.
• For $\epsilon = 0.05$, $\delta \leq 0.0125$.

Would you like more detailed steps or have any questions?

Here are 5 related questions to expand on the topic:

1. How does the formal definition of a limit differ from the intuitive definition?
2. Why is it important to express limits in terms of $\epsilon$ and $\delta$?
3. How can this process be applied to functions that are not linear?
4. How would you find $\delta$ if the limit expression had a higher degree polynomial?
5. Can $\delta$ ever be larger than the $\epsilon$ given?

Tip: When dealing with limits, always ensure to simplify the expression as much as possible before solving for $\delta$.