To find the limit $\lim_{x \to 5} f(x)$ using the provided graph, we need to observe the behavior of the function $f(x)$ as $x$ approaches 5 from both the left and the right.

1. Left-hand limit ($\lim_{x \to 5^-} f(x)$): This represents the value that $f(x)$ approaches as $x$ gets closer to 5 from values less than 5. By examining the graph, as $x$ approaches 5 from the left, the function $f(x)$ seems to approach a value of $-3$.

2. Right-hand limit ($\lim_{x \to 5^+} f(x)$): This represents the value that $f(x)$ approaches as $x$ gets closer to 5 from values greater than 5. From the graph, as $x$ approaches 5 from the right, $f(x)$ also approaches $-3$.

Since both the left-hand limit and the right-hand limit are equal and approach $-3$, we can conclude that the two-sided limit exists and is equal to $-3$.

Thus, the correct answer is:

$\lim_{x \to 5} f(x) = -3$

Would you like more details or have any questions about this explanation?

Here are 5 related questions that expand on this concept:

1. What does it mean if the left-hand limit and right-hand limit are not equal at a point?
2. How do you find a limit using a graph when the function has a hole or discontinuity?
3. Can a function be continuous at a point if the limit exists but does not equal the function's value at that point?
4. What are some common types of discontinuities that can appear in graphs?
5. How can you determine the behavior of a function as $x$ approaches infinity from its graph?

Tip: When evaluating limits graphically, pay close attention to the direction from which $x$ is approaching the point of interest. This can help identify any differences between left-hand and right-hand limits, revealing more about the function's behavior near that point.