The problem states that the function $f(x)$ satisfies the condition:

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

We are tasked with determining the largest value of $\delta > 0$ that satisfies the following conditions:

Part a.

If $0 < |x - 2| < \delta$, then $|f(x) - 2| < \frac{1}{2}$.

To solve this, we need to understand the definition of a limit and apply the epsilon-delta definition of limits.

Step 1: Define the Epsilon-Delta Condition

From the epsilon-delta definition of a limit:

$\lim_{x \to a} f(x) = L$

means that for every $\epsilon > 0$, there exists a $\delta > 0$ such that:

$0 < |x - a| < \delta \quad \Rightarrow \quad |f(x) - L| < \epsilon.$

In our case:

• $a = 2$
• $L = 2$
• $\epsilon = \frac{1}{2}$

So, we need to find the largest $\delta > 0$ such that for all $x$ satisfying $0 < |x - 2| < \delta$, it holds that:

$|f(x) - 2| < \frac{1}{2}.$

Step 2: Look at the Graph of $f(x)$

The graph of the function $f(x)$ is provided, and since the limit as $x \to 2$ is $2$, the function is approaching the value $2$ as $x$ approaches $2$. In order to determine $\delta$, you would examine the graph to identify how close $x$ needs to be to $2$ for the function to stay within $\frac{1}{2}$ of $2$.

Step 3: Determine the $\delta$ from the Graph

Based on the graph (which we don't have here), you can visually determine the $\delta$ that satisfies the condition. Find the $x$-values that are within a vertical distance of $\frac{1}{2}$ from $2$ on the $y$-axis, and identify the corresponding $x$-values on the horizontal axis near $2$. The distance between $x = 2$ and these $x$-values will give you $\delta$.

Part b.

The process for part b is the same, but this time you are asked to find $\delta$ such that:

$|f(x) - 2| < \frac{1}{4}.$

Repeat the process as described in part a but adjust the vertical tolerance to $\frac{1}{4}$ instead of $\frac{1}{2}$.

Summary of Steps

1. Look at the graph of $f(x)$ near $x = 2$.
2. For part a, find the largest $\delta$ such that $|f(x) - 2| < \frac{1}{2}$ for $0 < |x - 2| < \delta$.
3. For part b, find the largest $\delta$ such that $|f(x) - 2| < \frac{1}{4}$ for $0 < |x - 2| < \delta$.

If you'd like, you can upload the graph, and I can help with more detailed analysis based on the graph. Let me know!