Given the points \( (2.5, 6) \), \( (5, 0) \), and \( (7.5, 6) \) for the function \( f(x) \), we can interpret that \( f(x) \) follows some kind of pattern, possibly a quadratic or piecewise-defined function. Based on these points, let's analyze the behavior of \( f(x) \) as \( x \to \infty \).

### Step 1: Analyzing the Points
- At \( x = 2.5 \), \( f(x) = 6 \).
- At \( x = 5 \), \( f(x) = 0 \).
- At \( x = 7.5 \), \( f(x) = 6 \).

### Step 2: Estimating the Behavior of \( f(x) \)
These points suggest that \( f(x) \) could be a parabola that opens upwards or downwards, or it could be a piecewise function with symmetry around \( x = 5 \). However, the important observation is that as \( x \) increases beyond \( 7.5 \), no further points are provided, so we need to make assumptions about the behavior of \( f(x) \).

If we assume that \( f(x) \) does not grow indefinitely and instead returns to zero or stays bounded as \( x \to \infty \), then the limit \( \lim_{x \to \infty} f(x) \) could be \( 0 \).

### Step 3: Evaluating the Limit of \( g(x) \)
Given that \( g(x) = -9f(x) \), the behavior of \( g(x) \) will be directly related to the behavior of \( f(x) \).

#### Case 1: If \( \lim_{x \to \infty} f(x) = 0 \):
\[
\lim_{x \to \infty} g(x) = -9 \times 0 = 0
\]
In this case, the limit of \( g(x) \) as \( x \to \infty \) is \( 0 \).

### Conclusion:
The most reasonable assumption, based on the points given and typical behavior of functions, is that \( \lim_{x \to \infty} f(x) = 0 \). Therefore:

\[
\lim_{x \to \infty} g(x) = 0
\]

The correct answer is **0**.

Would you like further details or have any questions?

### Relative Questions:
1. Could the function \( f(x) \) be periodic beyond the given points?
2. Does \( f(x) \) have symmetry that could suggest a pattern?
3. What type of function could fit these points (e.g., quadratic, sinusoidal)?
4. Is there any information about the behavior of \( f(x) \) at negative infinity?
5. Could \( f(x) \) be a piecewise function with different behaviors in different regions?

### Tip:
When evaluating limits, especially with unknown functions, analyze given points carefully and consider common function behaviors, such as symmetry, periodicity, or boundedness.

Understand the behavior of a function based on given points (2.5, 6), (5, 0), and (7.5, 6). Analyze the limit as x approaches infinity and evaluate the related function g(x). Suitable for advanced high school students studying functions and limits.

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