To solve this problem, we need to analyze the graph of the function $g(x)$ based on the provided description and identify different scenarios for limits and continuity. Let's address each part step by step.

(a) $\lim_{x \to a} g(x)$ does not exist, but $g(a)$ is defined.

This condition describes a point where the function is discontinuous, but the function value is still defined. A common situation is when there is a jump discontinuity or a break in the graph.

From the description:

• At $x = 4$, the graph has an open point at $(4, 0.5)$ and a closed point at $(4, 2.25)$. This means that $g(4)$ is defined as 2.25, but the limit does not exist because the left-hand and right-hand limits do not match.

Thus: $a = 4$

(b) $\lim_{x \to a} g(x)$ exists, but $g(a)$ is not defined.

This condition indicates that the limit exists, but there is a hole (open point) in the graph at $x = a$.

From the description:

• At $x = 2$, there is a discontinuity where the graph has open points at both $(2, 2.75)$ and $(2, 1.75)$. So, the limit exists (since both sides converge to a value), but the function is not defined at $x = 2$.

Thus: $a = 2$

(c) $\lim_{x \to a^-} g(x)$ and $\lim_{x \to a^+} g(x)$ both exist, but $\lim_{x \to a} g(x)$ does not exist.

This condition describes a jump discontinuity where the left-hand and right-hand limits both exist but are not equal.

From the description:

• At $x = 4$, the left-hand limit approaches 2.25, and the right-hand limit approaches 0.5. Since these limits do not match, the overall limit does not exist.

Thus:

• Smaller value: $a = 4$

At $x = 2$, there is also a jump discontinuity, with left and right limits approaching 2.75 and 1.75, respectively. So the limit does not exist at $x = 2$ either.

Thus:

• Larger value: $a = 2$

(d) $\lim_{x \to a^+} g(x) = g(a)$ but $\lim_{x \to a^-} g(x) \neq g(a)$.

This condition describes a point where the right-hand limit equals the function value, but the left-hand limit does not.

From the description:

• At $x = 4$, the right-hand limit approaches 2.25, which is equal to the value of $g(4) = 2.25$, but the left-hand limit is 0.5, so they do not match.

Thus: $a = 4$

Final Answers:

• (a) $a = 4$
• (b) $a = 2$
• (c) Smaller value: $a = 4$, Larger value: $a = 2$
• (d) $a = 4$

Let me know if you'd like further details or have any questions!

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Here are 5 follow-up questions:

1. What is the difference between a jump discontinuity and a removable discontinuity?
2. How do we define a limit at a point when the function has a hole?
3. Why does the existence of a limit at a point not guarantee that the function is continuous there?
4. What are left-hand and right-hand limits, and how do they affect the overall limit at a point?
5. How can we tell if a function is defined at a point from its graph?

Tip: A function can have different types of discontinuities: jump, removable, and infinite, each affecting the limits in distinct ways.