To analyze the graph and find the limit of $\lim_{x \to 4} f(x)$, let's examine the behavior of the function $f(x)$ as $x$ approaches 4 from both the left and the right sides.

From the image, here's what we can observe:

• As $x$ approaches 4 from the left (i.e., $x \to 4^-$), the function $f(x)$ seems to decrease without bound, which means it tends to $-\infty$.
• As $x$ approaches 4 from the right (i.e., $x \to 4^+$), the function $f(x)$ increases without bound, which means it tends to $+\infty$.

Since the left-hand limit $\lim_{x \to 4^-} f(x) = -\infty$ and the right-hand limit $\lim_{x \to 4^+} f(x) = +\infty$ are not equal, we can conclude that:

$\lim_{x \to 4} f(x) \text{ does not exist}.$

Would you like more details on why this occurs? Here are some questions that can deepen your understanding:

1. What conditions must be met for a limit to exist at a specific point?
2. How do vertical asymptotes affect the behavior of limits?
3. What is the difference between one-sided and two-sided limits?
4. How does the concept of infinity play into the existence of limits?
5. Can a limit approach infinity and still be defined?

Tip: When examining graphs for limits, always check the behavior from both the left and the right sides of the point in question.