https://file.aiquickdraw.com/mathbot/file/1726788007199geu2wyme.jpg

The problem consists of two parts. Let's break down both parts:

### Part (a):
You are asked to sketch a graph for a function \( f(x) \) that satisfies several limits. Let's analyze them one by one:

1. \( \lim_{x \to 1^-} f(x) = \infty \): As \( x \) approaches 1 from the left, \( f(x) \) tends towards \( \infty \) (positive infinity).
2. \( \lim_{x \to 1^+} f(x) = -\infty \): As \( x \) approaches 1 from the right, \( f(x) \) tends towards \( -\infty \) (negative infinity).
3. \( \lim_{x \to 2^-} f(x) = -\infty \): As \( x \) approaches 2 from the left, \( f(x) \) tends towards \( -\infty \).
4. \( \lim_{x \to 2^+} f(x) = -\infty \): As \( x \) approaches 2 from the right, \( f(x) \) also tends towards \( -\infty \). This implies a vertical asymptote at \( x = 2 \) from both sides.
5. \( \lim_{x \to 4} f(x) = \infty \): As \( x \) approaches 4, \( f(x) \) tends towards \( \infty \).
6. \( \lim_{x \to \infty} f(x) = -\infty \): As \( x \) approaches infinity, \( f(x) \) tends towards \( -\infty \), indicating the behavior of the function far to the right.
7. \( \lim_{x \to -\infty} f(x) = 2 \): As \( x \) approaches negative infinity, \( f(x) \) approaches 2, suggesting a horizontal asymptote at \( y = 2 \).

### Part (b):
You are asked to determine several limits based on your graph:

- \( \lim_{x \to 1} f(x) \)
- \( \lim_{x \to 2} f(x) \)
- \( \lim_{x \to 4^-} f(x) \)
- \( \lim_{x \to 4^+} f(x) \)

These can be interpreted from the graph after sketching the behavior of the function according to the given limits.

Would you like me to generate a visual sketch of the graph based on these conditions?

Neatly sketch a graph of a function f(x) where all the following are true:
lim_{x → 1^-} f(x) = ∞,
lim_{x → 1^+} f(x) = -∞,
lim_{x → 2^-} f(x) = -∞,
lim_{x → 2^+} f(x) = -∞,
lim_{x → 4} f(x) = ∞,
lim_{x → ∞} f(x) = -∞,
lim_{x → -∞} f(x) = 2.

Sketch a graph based on limit behaviors including vertical and horizontal asymptotes at specific points. Analyze how f(x) behaves as x approaches certain values and infinity.

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