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Let's solve the given problem:

The function is \( f(x) = \frac{25x + 7}{\sqrt{5x^2 + 2}} \), and we are asked to find the limit as \( x \to \pm \infty \), which will help us determine whether the graph of \( f(x) \) has a horizontal asymptote.

### Step 1: Analyze the behavior as \( x \to \infty \)
As \( x \to \infty \), the dominant terms in the numerator and denominator are the highest powers of \( x \). Specifically:
- In the numerator, the dominant term is \( 25x \).
- In the denominator, \( \sqrt{5x^2 + 2} \) behaves like \( \sqrt{5x^2} = \sqrt{5} |x| = \sqrt{5} x \) as \( x \to \infty \).

Thus, we approximate the function for large \( x \):
\[
f(x) \approx \frac{25x}{\sqrt{5}x} = \frac{25}{\sqrt{5}} = 5\sqrt{5}.
\]

Hence, the limit as \( x \to \infty \) is:
\[
\lim_{x \to \infty} f(x) = 5\sqrt{5}.
\]

### Step 2: Analyze the behavior as \( x \to -\infty \)
As \( x \to -\infty \), we still have:
- In the numerator, the dominant term is \( 25x \).
- In the denominator, \( \sqrt{5x^2 + 2} \approx \sqrt{5} |x| = \sqrt{5} (-x) \).

Thus, for large negative \( x \):
\[
f(x) \approx \frac{25x}{\sqrt{5}(-x)} = -\frac{25}{\sqrt{5}} = -5\sqrt{5}.
\]

Hence, the limit as \( x \to -\infty \) is:
\[
\lim_{x \to -\infty} f(x) = -5\sqrt{5}.
\]

### Conclusion: Horizontal Asymptotes
Since the limits as \( x \to \infty \) and \( x \to -\infty \) are finite and nonzero, we conclude that the graph of \( f(x) \) has two horizontal asymptotes:
- \( y = 5\sqrt{5} \) as \( x \to \infty \),
- \( y = -5\sqrt{5} \) as \( x \to -\infty \).

Would you like any additional details on the steps? Here are five related questions to explore:

1. What does it mean for a function to have a horizontal asymptote?
2. How do you determine the end behavior of more complex functions like rational or radical functions?
3. Why do we ignore the lower-order terms when calculating limits at infinity?
4. Can a function have more than one horizontal asymptote?
5. How do vertical asymptotes differ from horizontal asymptotes?

**Tip:** When calculating limits at infinity, focus on the highest-degree terms, as they dominate the function's behavior for large values of \( x \).

Find lim x → ±∞ f(x) for f(x) = (25x + 7)/√(5x² + 2). Determine if the graph of f has a horizontal asymptote.

Find the limit of f(x) = (25x + 7)/√(5x² + 2) as x approaches ±∞, and determine if the graph of f has a horizontal asymptote. This problem covers key concepts such as limits at infinity and asymptotic behavior of radical functions. Suitable for Grades 11-12.

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