The problem provides several key limits and function values for $f(x)$. Let's break it down:

1. Behavior around $x = 2$:

   • $\lim_{x \to 2^-} f(x) = -\infty$ indicates a vertical asymptote at $x = 2$, where the function goes to $-\infty$ as $x$ approaches 2 from the left.
   • $\lim_{x \to 2^+} f(x) = \infty$ indicates the function goes to $\infty$ as $x$ approaches 2 from the right. So, there's a jump discontinuity at $x = 2$.

2. Behavior as $x \to -\infty$:

   • $\lim_{x \to -\infty} f(x) = 2$ implies there is a horizontal asymptote at $y = 2$ as $x \to -\infty$.

3. Behavior as $x \to \infty$:

   • $\lim_{x \to \infty} f(x) = 2$ suggests another horizontal asymptote at $y = 2$ as $x \to \infty$.

4. Behavior around $x = 3$:

   • $\lim_{x \to 3^-} f(x) = 5$ and $\lim_{x \to 3^+} f(x) = 5$, which means the function approaches 5 from both sides of $x = 3$, but $f(3)$ is undefined. This implies a hole at $x = 3$.

5. Function values:

   • $f(1) = 0$ gives a point at $(1, 0)$, which is an x-intercept.
   • $f(0) = 1$ gives another point at $(0, 1)$.

Sketching the Graph

Based on this information, the graph would exhibit the following features:

• A vertical asymptote at $x = 2$.
• A horizontal asymptote at $y = 2$ for both $x \to -\infty$ and $x \to \infty$.
• A hole at $x = 3$, $y = 5$.
• Intercepts at $(1, 0)$ and $(0, 1)$.

Would you like more details on the sketching process, or clarification on specific parts?

Here are five related questions:

1. How do vertical asymptotes influence the graph of a function?
2. What is the significance of horizontal asymptotes in behavior as $x \to \infty$ and $x \to -\infty$?
3. How do you interpret jump discontinuities like the one at $x = 2$?
4. What is the difference between a removable discontinuity (like the hole at $x = 3$) and a non-removable one?
5. How does knowing $f(1) = 0$ and $f(0) = 1$ help in plotting points?

Tip: When sketching graphs of functions, always start with identifying the behavior at critical points like asymptotes, intercepts, and discontinuities.