https://file.aiquickdraw.com/mathbot/file/1729832151251d4t8kwh0.jpg

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.

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### (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.

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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.

Determine for which values of x = a the limit of f(x) exists but f is not continuous at x = a. Additionally, determine for which values of x = a the function is continuous but not differentiable at x = a.

This math problem involves analyzing the continuity and differentiability of a function using its graph. It covers core concepts such as limits, continuity, and differentiability at specific points. Suitable for students in Grades 11-12, the problem tests the understanding of sharp corners and discontinuities.

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