Let's analyze the problem based on the image you have provided.

We are asked to determine the limits of the function $f(x)$ for different values of $x$, using the graph.

Here's a breakdown of the graph:

• The graph of the function is given with specific behavior as $x$ approaches certain values. Based on the instructions, the limits are:

(a) $\lim_{x \to -8} f(x)$

• Looking at the graph near $x = -8$, there is a hole in the graph, meaning that the function approaches a specific value from both sides but does not actually reach it. From the image, it appears that as $x$ approaches $-8$, the $y$-value approaches $-2$. Therefore, $\lim_{x \to -8} f(x) = -2$.

(b) $\lim_{x \to 2^-} f(x)$

• For the left-hand limit as $x$ approaches 2, look at the graph as $x$ approaches 2 from values less than 2. The graph seems to approach $y = 0$ from the left. So, $\lim_{x \to 2^-} f(x) = 0$.

(c) $\lim_{x \to 2^+} f(x)$

• For the right-hand limit as $x$ approaches 2, observe how the function behaves from the right of $x = 2$. The function appears to approach 2, but there's no continuity here. Hence, this limit might not exist. The correct answer is likely "none" for $\lim_{x \to 2^+} f(x)$.

(d) $\lim_{x \to \infty} f(x)$

• As $x \to \infty$, the graph shows that the function increases and heads towards $y = 18$, so the correct value for this limit is 18.

Final answers:

• (a) $\lim_{x \to -8} f(x) = -2$
• (b) $\lim_{x \to 2^-} f(x) = 0$
• (c) $\lim_{x \to 2^+} f(x) = \text{none}$
• (d) $\lim_{x \to \infty} f(x) = 18$ (correct)

Let me know if you'd like more details or have questions about this problem.

Related Questions:

1. What are left-hand and right-hand limits, and why are they important?
2. How do you interpret a limit when there's a hole in the graph at a specific point?
3. What does it mean when a limit "does not exist"?
4. How does behavior at infinity affect the interpretation of limits?
5. What are the differences between continuous and discontinuous functions in limit calculations?

Tip:

Always check for continuity and jumps in the graph at the point you're evaluating for the limit. These give clues about whether the limit exists or not!