To solve this problem, we need to find the limit of the function $f(x)$ as $x$ approaches $4$ based on the graph provided. Let's analyze the graph:

1. As $x$ approaches $4$ from the left ($x \to 4^-$), the values of $f(x)$ appear to get closer to $2$.
2. As $x$ approaches $4$ from the right ($x \to 4^+$), the values of $f(x)$ also approach $2$.

Since the left-hand limit and the right-hand limit are equal: $\lim_{x \to 4} f(x) = 2$

So, the answer is: $\lim_{x \to 4} f(x) = 2$

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

Here are 5 related questions to expand your understanding:

1. What is the definition of a limit in calculus?
2. How can we determine if a function is continuous at a point using limits?
3. What happens if the left-hand and right-hand limits are not equal?
4. How do you find the limit of a piecewise function at the point where the definition changes?
5. What is the significance of open and closed circles on a graph when evaluating limits?

Tip: When evaluating limits from a graph, always check both the left-hand and right-hand behavior of the function as it approaches the point in question.