The image contains two graphs labeled (A) and (B) that depict the function $y = f(x)$, each showing how close the function is to a certain limit. The task is to find values for $c$, $L$, $\epsilon$, and $\delta$ such that the following holds:

$\text{If } 0 < |x - c| < \delta, \text{ then } |f(x) - L| < \epsilon.$

This is the formal definition of the limit $\lim_{x \to c} f(x) = L$, with $\epsilon$ and $\delta$ representing how close the function values are to the limit $L$ when $x$ is within $\delta$ of $c$.

For Graph (A):

• The function $f(x)$ approaches $L = 4$ as $x \to 0$.
• You can choose $c = 0$, since the vertical line is around $x = 0$.
• The values of $\epsilon$ can be chosen based on the vertical distance between $f(x)$ and $L = 4$. From the graph, $f(x)$ stays between 3.5 and 4.8, so an appropriate $\epsilon$ could be 0.5.
• For $\delta$, the horizontal distance between $x = 0$ and the points where $f(x)$ starts to deviate from the bounds of $\epsilon$ gives $\delta \approx 0.1$, since the graph remains within $|f(x) - L| < \epsilon$ for $-0.1 < x < 0.1$.

For Graph (B):

• The function $f(x)$ approaches $L = 10$ as $x \to 3$.
• You can choose $c = 3$, since the vertical line is around $x = 3$.
• The values of $\epsilon$ can be chosen based on the vertical distance between $f(x)$ and $L = 10$. From the graph, $f(x)$ stays between 9.8 and 10.4, so an appropriate $\epsilon$ could be 0.2.
• For $\delta$, the horizontal distance between $x = 3$ and the points where $f(x)$ starts to deviate from the bounds of $\epsilon$ gives $\delta \approx 0.1$, since the graph remains within $|f(x) - L| < \epsilon$ for $2.9 < x < 3.1$.

Would you like more detailed explanations or have any specific questions about this?

Here are 5 follow-up questions:

1. What does the formal definition of a limit represent graphically?
2. How would you select different values for $\delta$ if the graph changed?
3. What happens when $|x - c|$ becomes greater than $\delta$?
4. Can $\epsilon$ be smaller than the chosen values in the graphs? Why or why not?
5. How do you interpret $f(x)$ approaching $L$ as $x \to c$?

Tip: When choosing $\epsilon$ and $\delta$, visualize the vertical and horizontal distances that limit how far $f(x)$ can vary from the limit $L$.