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From the graph of the function \( f(x) \) that you provided, we are tasked with finding the values of \( a \) where \( \lim_{x \to a} f(x) \) does not exist.

To identify these values, we need to examine points where the behavior of the function is discontinuous or undefined. These cases typically occur at:

1. **Vertical asymptotes**: The function tends to infinity on either side.
2. **Jumps**: The left-hand and right-hand limits are different.
3. **Oscillations**: The function fluctuates indefinitely.

From the graph:
- There is a **vertical asymptote** at \( x = 1 \) because the function spikes upwards as \( x \) approaches 1 from either side.
- There also appears to be a **jump discontinuity** at \( x = 2 \), as the function value jumps to a new point (marked by an open dot and a separate point).

Thus, the limit \( \lim_{x \to a} f(x) \) does not exist at \( x = 1 \) (due to the vertical asymptote) and at \( x = 2 \) (due to the jump discontinuity).

The answer is:

\[
a = 1, 2
\]

Would you like more details on the types of discontinuities or how limits behave in these cases?

Here are some follow-up questions to expand on this:
1. What are the different types of discontinuities that can cause limits not to exist?
2. How can you formally determine a vertical asymptote from a graph?
3. Can a function have an infinite limit and still be continuous?
4. What is the difference between one-sided limits and two-sided limits?
5. How can you identify a removable discontinuity from a graph?

**Tip:** Always check both sides of the limit (left-hand and right-hand) to confirm if the limit exists or differs at a point.

At what numbers a does lim(x → a) f(x) not exist?

Analyze the graph of a function to find where the limit does not exist. This problem covers key calculus concepts, including limits, discontinuities, vertical asymptotes, and jump discontinuities. Suitable for Grades 11-12 and college-level students.

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