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

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 $x$.