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.