To sketch a graph of a function $f(x)$ that satisfies the given conditions, let's break down each condition and what it implies for the behavior of the function:

1. $f(0) = 0$: The function passes through the origin at the point $(0, 0)$.

2. $\lim_{x \to 2^-} f(x) = \infty$: As $x$ approaches 2 from the left, the function approaches positive infinity. This indicates a vertical asymptote at $x = 2$, with the graph increasing without bound on the left side of $x = 2$.

3. $\lim_{x \to 2^+} f(x) = -\infty$: As $x$ approaches 2 from the right, the function approaches negative infinity. This reinforces the vertical asymptote at $x = 2$, with the graph decreasing without bound on the right side of $x = 2$.

4. $\lim_{x \to -\infty} f(x) = -4$: As $x$ approaches negative infinity, the function levels off and approaches $y = -4$, indicating a horizontal asymptote at $y = -4$ on the left side.

5. $\lim_{x \to \infty} f(x) = -4$: As $x$ approaches positive infinity, the function levels off and approaches $y = -4$, indicating a horizontal asymptote at $y = -4$ on the right side.

Key Features of the Graph:

• The graph will cross the origin at $(0, 0)$.
• It has a vertical asymptote at $x = 2$, with the function shooting up to infinity from the left side and down to negative infinity from the right side.
• There is a horizontal asymptote at $y = -4$ for both $x \to -\infty$ and $x \to \infty$, so the graph will level off at $y = -4$ far to the left and right.

Visual Description of the Graph:

• As $x \to -\infty$, the function will start near $y = -4$, slowly approaching $y = -4$ as a horizontal asymptote.
• As $x$ increases from negative values, the function will rise from $y = -4$, pass through the origin at $(0, 0)$, and increase toward positive infinity as $x \to 2^-$ (left of $x = 2$).
• After crossing the vertical asymptote at $x = 2$, the function will decrease sharply to negative infinity as $x \to 2^+$ (right of $x = 2$).
• After the sharp decrease at $x = 2$, the function will begin leveling off again as $x \to \infty$, approaching $y = -4$ as a horizontal asymptote.

Would you like me to generate a visual sketch of this graph? Let me know if you need more details!

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Here are some related questions for you:

1. How does the vertical asymptote at $x = 2$ affect the overall behavior of the function?
2. What happens to the function as $x \to -\infty$ and $x \to \infty$?
3. Can a function have more than one horizontal asymptote? Why or why not?
4. How would the graph change if one of the limits at $x = \infty$ or $x = -\infty$ was different?
5. What types of functions exhibit both vertical and horizontal asymptotes?

Tip: Always check the behavior of the function around asymptotes and intercepts to better understand the graph’s structure!