Sketch the graph of an example of a function f that satisfies all of the given conditions.
f(0) = 0,   
lim 
x→2− 
(f(x)) = ∞,   
lim 
x→2+ 
(f(x)) = −∞,   
lim 
x→−∞ 
(f(x)) = −4,   
lim 
x→∞ 
(f(x)) = −4

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!

Sketch the graph of an example of a function f that satisfies all of the given conditions. f(0) = 0, lim x→2− (f(x)) = ∞, lim x→2+ (f(x)) = −∞, lim x→−∞ (f(x)) = −4, lim x→∞ (f(x)) = −4.

Learn how to sketch the graph of a function f(x) given specific conditions on limits, vertical asymptotes at x=2, and horizontal asymptotes at y=-4. This solution explores key calculus concepts like limit behavior and continuity, suitable for undergraduate calculus students.

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