Looking at the graph, let's address the two parts of the question:

(a) Limit Exists but Function is Not Continuous

For this part, we need to find a point where the limit of $f(x)$ exists, but $f(x)$ is not continuous at $x = a$. This happens when the left-hand and right-hand limits of $f(x)$ as $x \to a$ are equal, but $f(a)$ is not defined or differs from the limit.

In the graph:

• At $x = 2$, the limit exists from both sides, but there is a hole at $x = 2$, meaning $f(2)$ is not continuous.

This is why the correct answer to (a) is $x = 2$, which you correctly answered.

(b) Function is Continuous but Not Differentiable

For this part, we are looking for a point where the function is continuous but has a sharp corner, cusp, or vertical tangent line, making it non-differentiable at $x = a$.

From the graph:

• At $x = 1$, the function is continuous, but there is a sharp corner, indicating that $f$ is not differentiable at this point. However, $x = 1$ seems like it should be correct, but it was marked wrong.

It is possible that there is another point where this condition holds. I would suggest looking at $x = 3$, where the function has a sharp corner as well. Thus, $x = 3$ might be the correct answer instead.

Do you need further clarification or details? Here are 5 related questions for deeper understanding:

1. Why is a function not differentiable at a sharp corner?
2. How can we verify the continuity of a function at a point?
3. What is the relationship between differentiability and continuity?
4. Can you identify other points on the graph where the function might be discontinuous?
5. What is the significance of a limit existing but the function not being continuous?

Tip: When analyzing continuity and differentiability, always check for gaps, jumps, or sharp turns in the graph.