Let's analyze the graph to answer the questions based on the provided descriptions:

1. (a) $\lim\limits_{x \to a} g(x)$ does not exist but $g(a)$ is defined.

   • For this to be true, there must be a discontinuity where the left-hand limit and the right-hand limit are different, but the function value at $a$ exists.
   • Observing the graph, at $x = 4$, the left-hand limit and right-hand limit of $g(x)$ do not match, but $g(4)$ is defined.

   Answer: $a = 4$

2. (b) $\lim\limits_{x \to a} g(x)$ exists but $g(a)$ is not defined.

   • For this to be true, the function should have a hole at $x = a$, meaning that the limit exists but the actual value $g(a)$ is undefined.
   • This occurs at $x = 2$, where the limit exists but $g(2)$ is not defined as there's a hole.

   Answer: $a = 2$

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

   • This occurs where the left-hand limit and right-hand limit both exist but are not equal.
   • As seen in the graph, at $x = 4$, the left-hand limit is different from the right-hand limit.

   Smaller value $a = 4^-$
   Larger value $a = 4^+$

4. (d) $\lim\limits_{x \to a} g(x) = g(a)$ but $\lim\limits_{x \to a^+} g(x) \neq g(a)$.

   • For this to be true, the left-hand limit and function value should match, but the right-hand limit does not match the function value at $x = a$.
   • This is tricky, but the point $x = 5$ appears to be the correct point based on the graph.

   Answer: $a = 5$

Would you like more details on these answers or have any other questions?

Here are five related questions you might consider:

1. What is the difference between one-sided limits and two-sided limits?
2. How does the existence of limits relate to continuity?
3. What types of discontinuities can exist in a function?
4. How can you algebraically determine if a function has a removable discontinuity?
5. What are some practical applications of limits in real-world scenarios?

Tip: When analyzing limits graphically, always check the behavior from both sides of the point in question to see if they match. This will help in identifying discontinuities and the type of limit that exists.