From the graph provided, we need to evaluate the following limits:

a. $\lim\limits_{x \to 3} f(x)$

• As $x$ approaches 3 from both the left and the right sides, the value of $f(x)$ is approaching 8 (since the function appears continuous around $x = 3$, and the $y$-value at $x = 3$ is 8).
• Therefore, $\lim\limits_{x \to 3} f(x) = 8$.

b. $\lim\limits_{x \to 6} f(x)$

• As $x$ approaches 6 from both the left and the right sides, the value of $f(x)$ is approaching 5 (again, the graph shows that the function is continuous at $x = 6$, and the $y$-value at $x = 6$ is 5).
• Therefore, $\lim\limits_{x \to 6} f(x) = 5$.

Final answers:

• a. $\lim\limits_{x \to 3} f(x) = 8$
• b. $\lim\limits_{x \to 6} f(x) = 5$

Would you like further explanations or have any questions? Here are 5 questions that expand on this concept:

1. What happens to a limit if the function is discontinuous at that point?
2. How can you identify if a function has a limit at a given point using a graph?
3. What is the difference between a one-sided limit and a two-sided limit?
4. Can limits exist even if the function is undefined at the point of interest?
5. How would you calculate limits algebraically instead of graphically?

Tip: When dealing with limits graphically, check both the left and right side of the point. If they converge to the same value, the limit exists!