Let's analyze the provided graph and answer the given questions:

1. a. Find the limit $\lim_{{x \to 2^-}} f(x)$:

   • This asks for the limit as $x$ approaches 2 from the left (i.e., from values less than 2). Looking at the graph near $x = 2$ from the left-hand side, the curve seems to approach a value around 1.

   • So, $\lim_{{x \to 2^-}} f(x) = 1$.

2. b. Find the limit $\lim_{{x \to 2^+}} f(x)$:

   • This asks for the limit as $x$ approaches 2 from the right (i.e., from values greater than 2). Looking at the graph near $x = 2$ from the right-hand side, the curve seems to approach a value around 3.

   • So, $\lim_{{x \to 2^+}} f(x) = 3$.

3. c. Find the limit $\lim_{{x \to 2}} f(x)$:

   • The overall limit as $x$ approaches 2 depends on whether the left-hand limit equals the right-hand limit. Since $\lim_{{x \to 2^-}} f(x) = 1$ and $\lim_{{x \to 2^+}} f(x) = 3$, the two one-sided limits are not equal.

   • Therefore, $\lim_{{x \to 2}} f(x)$ does not exist.

4. d. Find the function value $f(2)$:

   • The graph shows a filled dot at $x = 2$ with a corresponding $y$-value of 2. This indicates that $f(2) = 2$.

   • So, $f(2) = 2$.

Would you like more details or clarifications on any part of this explanation?

Here are 5 additional related questions to deepen your understanding:

1. What is the difference between a one-sided limit and a two-sided limit?
2. How do you determine when a limit does not exist?
3. Why is the value of the function $f(2)$ not necessarily the same as $\lim_{{x \to 2}} f(x)$?
4. How can you visually identify discontinuities in a graph?
5. What types of discontinuities exist and how are they classified?

Tip: Always check for one-sided limits when evaluating a two-sided limit, as discrepancies can indicate the limit does not exist.