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The function given is:

\[
f(x) = \frac{7x - 5}{\sqrt{16x^2 + 3}}
\]

We are asked to find the horizontal asymptote(s), if any. Horizontal asymptotes are determined by analyzing the behavior of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \).

### Step-by-Step Process:

1. **As \( x \to \infty \):**
   - The dominant term in the numerator is \( 7x \).
   - In the denominator, as \( x \to \infty \), \( \sqrt{16x^2 + 3} \approx \sqrt{16x^2} = 4|x| = 4x \) (since \( x \) is positive).
   - So, the function for large \( x \) behaves like:

   \[
   f(x) \approx \frac{7x}{4x} = \frac{7}{4}
   \]

   Therefore, as \( x \to \infty \), \( f(x) \to \frac{7}{4} \).

2. **As \( x \to -\infty \):**
   - The dominant term in the numerator is still \( 7x \).
   - In the denominator, \( \sqrt{16x^2 + 3} \approx \sqrt{16x^2} = 4|x| = 4(-x) = -4x \) (since \( x \) is negative).
   - So, the function for large negative \( x \) behaves like:

   \[
   f(x) \approx \frac{7x}{-4x} = -\frac{7}{4}
   \]

   Therefore, as \( x \to -\infty \), \( f(x) \to -\frac{7}{4} \).

### Conclusion:

The function has two horizontal asymptotes:
- As \( x \to \infty \), \( f(x) \to \frac{7}{4} \).
- As \( x \to -\infty \), \( f(x) \to -\frac{7}{4} \).

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Would you like more details on any of the steps, or have any questions? 

Here are 5 related questions for further exploration:
1. How do you determine vertical asymptotes for a rational function?
2. How does the degree of polynomials in a function affect horizontal asymptotes?
3. What are the conditions for a function to have no horizontal asymptotes?
4. Can a function have both horizontal and slant asymptotes?
5. How would the behavior of the function change if the denominator had different powers of \(x\)?

**Tip:** Horizontal asymptotes describe the end behavior of a function, not necessarily how the function behaves for finite values of \(x\).

Find the horizontal asymptote(s), if any, of the function f(x) = (7x - 5) / sqrt(16x^2 + 3).

Learn how to find the horizontal asymptotes of the function f(x) = (7x - 5) / sqrt(16x^2 + 3). This problem explores the concepts of limits and asymptotes in rational functions, suitable for students in Grades 11-12.

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