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To find the horizontal asymptotes of the given function, we need to compute the limits as \(x\) approaches both positive infinity (\(x \to \infty\)) and negative infinity (\(x \to -\infty\)).

### Given function:
\[
f(x) = \frac{\sqrt{x^2 + 4x}}{7 - 7x}
\]

### Step-by-step solution:

#### 1. Limit as \(x \to \infty\):
We start by dividing the numerator and the denominator by \(x\), the highest power of \(x\) under the square root in the numerator and in the denominator.

\[
\lim_{x \to \infty} \frac{\sqrt{x^2 + 4x}}{7 - 7x}
\]

\[
= \lim_{x \to \infty} \frac{\sqrt{x^2(1 + \frac{4}{x})}}{7(1 - x)}
\]

\[
= \lim_{x \to \infty} \frac{|x|\sqrt{1 + \frac{4}{x}}}{7(1 - x)}
\]

As \(x \to \infty\), \(\frac{4}{x} \to 0\), so the square root simplifies:

\[
= \frac{|x| \cdot 1}{-7x} = -\frac{1}{7}
\]

Thus, the horizontal asymptote as \(x \to \infty\) is:
\[
y = -\frac{1}{7}
\]

#### 2. Limit as \(x \to -\infty\):
We apply the same method for \(x \to -\infty\):

\[
\lim_{x \to -\infty} \frac{\sqrt{x^2 + 4x}}{7 - 7x}
\]

Following the same process, divide by \(x\) (the highest power):

\[
= \lim_{x \to -\infty} \frac{|x|\sqrt{1 + \frac{4}{x}}}{7(1 - x)}
\]

Now, as \(x \to -\infty\), \(\frac{4}{x} \to 0\), and the absolute value of \(x\) is \(-x\) since \(x\) is negative.

\[
= \frac{-x}{7x} = \frac{1}{7}
\]

Thus, the horizontal asymptote as \(x \to -\infty\) is:
\[
y = \frac{1}{7}
\]

### Final Answer:
- Horizontal asymptote as \(x \to \infty\): \(y = -\frac{1}{7}\)
- Horizontal asymptote as \(x \to -\infty\): \(y = \frac{1}{7}\)

Would you like further clarification or any additional details?

Here are 5 related questions you might find useful:
1. How do we determine the horizontal asymptotes of more complex rational functions?
2. What is the significance of the absolute value in the square root simplification for limits?
3. How does dividing by the highest power of \(x\) help in finding limits at infinity?
4. Can a function have more than two horizontal asymptotes?
5. What are the key differences between vertical and horizontal asymptotes?

**Tip:** Always remember that when dealing with square roots in limits at infinity, the absolute value of \(x\) is crucial for correctly simplifying the expressions.

Find the horizontal asymptotes of the function \(f(x) = \frac{\sqrt{x^2 + 4x}}{7 - 7x}\).

This problem demonstrates how to find the horizontal asymptotes of the function f(x) = sqrt(x^2 + 4x) / (7 - 7x) by calculating the limits as x approaches infinity and negative infinity. Includes step-by-step explanations for simplifying limits using absolute values and factoring.

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