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The problem asks us to find the limit of the function \( C(x) = \frac{6x + 129}{x} \) as \( x \to \infty \), where \( x \) represents the number of products, and \( C(x) \) represents the average cost in dollars.

To find the limit, let's rewrite the function:

\[
C(x) = \frac{6x + 129}{x} = \frac{6x}{x} + \frac{129}{x} = 6 + \frac{129}{x}
\]

Now, as \( x \to \infty \), the term \( \frac{129}{x} \to 0 \) because the denominator grows infinitely large.

Thus:

\[
\lim_{x \to \infty} C(x) = 6 + 0 = 6
\]

So, the correct answer is:

- \(\lim_{x \to \infty} C(x) = 6\)

This corresponds to the fourth option in the list provided in the image.

### Would you like more details or have any questions?

Here are 5 related questions you might find helpful:

1. What does the term "limit" mean in calculus?
2. How do we find limits of more complex rational functions as \( x \to \infty \)?
3. Why does the term \( \frac{129}{x} \) approach 0 as \( x \to \infty \)?
4. How would the answer change if the function was \( C(x) = \frac{6x + 129}{x^2} \)?
5. What is the importance of limits in understanding the behavior of functions?

**Tip:** When finding the limit of a rational function as \( x \to \infty \), dividing each term by the highest power of \( x \) in the denominator can simplify the process!

The average cost for producing a product is given by the function C(x) = (6x + 129) / x for x > 0. Which of the following statements describes the cost as the number of products increases?

Explore how to find the limit of the average cost function C(x) = (6x + 129) / x as the number of products increases. This calculus problem illustrates rational functions and their behavior at infinity, suitable for advanced high school or introductory college students.

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