What are the horizontal asymptotes of 𝑓?of f?

(Give your answer as a comma‑separated list of equations. Express numbers in exact form. Use symbolic notation and fractions where needed.)
horizontal asymptote(s): (16x+1)/(\sqrt(4x^(2)+9))

To find the horizontal asymptotes of the function 

\[
f(x) = \frac{16x+1}{\sqrt{4x^2 + 9}},
\]

we need to analyze the behavior of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \).

### Step 1: Examine the behavior as \( x \to \infty \)
When \( x \to \infty \), the dominant terms in the numerator and denominator will control the behavior of the function. Let's analyze each part:

- The numerator \( 16x + 1 \) behaves like \( 16x \) for large values of \( x \).
- The denominator \( \sqrt{4x^2 + 9} \) behaves like \( \sqrt{4x^2} = 2x \) for large values of \( x \), because the constant term \( 9 \) becomes negligible compared to \( 4x^2 \).

Thus, for large \( x \),
\[
f(x) \approx \frac{16x}{2x} = 8.
\]
Therefore, as \( x \to \infty \), \( f(x) \to 8 \).

### Step 2: Examine the behavior as \( x \to -\infty \)
Similarly, as \( x \to -\infty \):

- The numerator \( 16x + 1 \) behaves like \( 16x \), which approaches a large negative value.
- The denominator \( \sqrt{4x^2 + 9} \) behaves like \( \sqrt{4x^2} = 2|x| \). For negative values of \( x \), this simplifies to \( 2(-x) = -2x \).

Thus, for large negative \( x \),
\[
f(x) \approx \frac{16x}{-2x} = -8.
\]
Therefore, as \( x \to -\infty \), \( f(x) \to -8 \).

### Conclusion: 
The function has two horizontal asymptotes:

\[
y = 8 \quad \text{and} \quad y = -8.
\]

### Final Answer:
The horizontal asymptotes are \( y = 8 \) and \( y = -8 \).

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Would you like further details on how asymptotes are derived or have any specific questions?

Here are 5 related questions to deepen your understanding:

1. How do vertical asymptotes differ from horizontal asymptotes?
2. Can a function have more than two horizontal asymptotes?
3. How do slant asymptotes form for rational functions?
4. How do we determine the behavior of a function near an asymptote?
5. How would you calculate the limits of more complex functions with roots?

**Tip**: The horizontal asymptotes represent the end behavior of the function, showing how it behaves as \( x \) grows very large or very small.

What are the horizontal asymptotes of f?
(Give your answer as a comma‑separated list of equations. Express numbers in exact form. Use symbolic notation and fractions where needed.)

horizontal asymptote(s): (16x+1)/sqrt(4x^2+9)

Explore how to determine the horizontal asymptotes of the function f(x) = (16x+1)/sqrt(4x^2+9). This solution explains the behavior of the function as x approaches infinity and negative infinity, showing that the asymptotes are y = 8 and y = -8. Perfect for high school students in Grades 10-12.

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