Let's analyze the given properties step by step in order to sketch the graph of the function $f(x)$:

1. As $x \to -\infty$, $f(x) \to 2$:

• This means that as $x$ moves towards negative infinity, the function approaches the value 2. So, the left tail of the graph will flatten out toward $y = 2$.

2. $\lim_{x \to 3^-} f(x) = -\infty$:

• As $x$ approaches 3 from the left, the function goes to negative infinity. This indicates a vertical asymptote at $x = 3$, where the function plunges downwards as it gets closer to 3 from the left side.

3. $\lim_{x \to 3^+} f(x) = \infty$:

• As $x$ approaches 3 from the right, the function shoots up towards positive infinity. This means that after crossing $x = 3$, the function ascends rapidly, reinforcing the vertical asymptote at $x = 3$.

4. As $x \to \infty$, $f(x) \to 2$:

• As $x$ moves towards positive infinity, the function approaches the value 2 again. So, the right tail of the graph will also flatten out towards $y = 2$.

5. $f(0) = -\frac{1}{3}$:

• At $x = 0$, the function value is $-\frac{1}{3}$. This gives a specific point on the graph: $(0, -\frac{1}{3})$.

Summary of the Graph:

• Horizontal asymptotes: At $y = 2$ as $x \to \pm\infty$.
• Vertical asymptote: At $x = 3$, with $f(x) \to -\infty$ as $x \to 3^-$ and $f(x) \to \infty$ as $x \to 3^+$.
• Point: $(0, -\frac{1}{3})$ is on the graph.

The graph will likely approach the horizontal asymptote $y = 2$ at both extreme ends (negative and positive infinity), and there will be a sharp drop and rise near $x = 3$ due to the vertical asymptote.

Would you like me to draw this graph for you?

Five related questions:

1. What is the significance of horizontal asymptotes in graphing functions?
2. How can vertical asymptotes affect the behavior of a graph?
3. What are the general steps to find limits of a function as $x \to \infty$?
4. How do you determine if a function has a jump discontinuity or infinite discontinuity?
5. Can a function have both vertical and horizontal asymptotes at the same point?

Tip:

When sketching a function with asymptotes, it's useful to mark the asymptotes first and then sketch the behavior of the graph around them.