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The graph provided shows a function \( f(x) \) with discontinuities and possible points where the derivative does not exist. Let's analyze the graph to answer the questions:

### (a) At what numbers \( a \) does \( \lim_{x \to a} f'(x) \) not exist?
To find where the derivative \( f'(x) \) does not exist, we look for points on the graph where:
- The function has a sharp corner or cusp.
- The tangent line is vertical (which means the slope is undefined).

From the graph:
1. At \( x = 1 \), there is a sharp corner.
2. At \( x = 3 \), there is a vertical tangent.

Therefore, the derivative does not exist at \( x = 1 \) and \( x = 3 \).

#### Answer for (a):  
\( a = 1, 3 \)

### (b) At what numbers \( a \) is \( f \) not continuous?
A function is not continuous at points where there are jumps, holes, or vertical asymptotes.

From the graph:
1. At \( x = 2 \), there is a jump discontinuity.

#### Answer for (b):  
\( a = 2 \)

### (c) At what numbers \( a \) does \( \lim_{x \to a} f'(x) \) exist but \( f \) is not continuous at \( a \)?
For this question, we need to find a point where the limit of the derivative exists but the function itself is not continuous. This is generally impossible because if a function is not continuous at a point, the derivative at that point also does not exist in the usual sense.

From the graph:
- There are no points where the limit of the derivative exists while the function is not continuous.

#### Answer for (c):  
\( a = \) (none)

### Summary of Answers:
(a) \( a = 1, 3 \)  
(b) \( a = 2 \)  
(c) \( a = \) (none)

Would you like a more detailed explanation of any part? Here are 5 related questions to explore further:

1. What are the conditions for a function to be differentiable at a point?
2. How can you determine if a limit exists using a graph?
3. What types of discontinuities are there, and how do they differ?
4. What is the relationship between continuity and differentiability?
5. How can we compute derivatives graphically without a formula?

**Tip:** To determine if a function is continuous at a point, check that the left-hand limit, right-hand limit, and the value of the function at that point are all equal.

The graph of a function f is given. Answer the following questions based on the graph:
(a) At what numbers does lim x→a f'(x) not exist?
(b) At what numbers is f not continuous?
(c) At what numbers does lim x→a f'(x) exist but f is not continuous at a?

Learn how to analyze continuity and differentiability from a given graph. This problem explores where the derivative does not exist, points of discontinuity, and the relationship between continuity and differentiability. Suitable for high school students studying calculus.

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