Your friend is calculating the horizontal asymptotes of a function, $f(x)$, by taking the limit of the expression as $x \to \infty$ and $x \to -\infty$. Here's a step-by-step analysis of their work:

1. The function to analyze:
   $f(x) = \frac{2.5x + 7}{\sqrt{5x^2 + 22}}$.

2. Taking the limit as $x \to \infty$: They begin with: $\lim_{x \to \infty} \frac{2.5x + 7}{\sqrt{5x^2 + 22}}.$

   • They divide the numerator and the denominator by $x$, yielding: $\frac{2.5 + \frac{7}{x}}{\sqrt{5 + \frac{22}{x^2}}}.$
   • As $x \to \infty$, both $\frac{7}{x} \to 0$ and $\frac{22}{x^2} \to 0$, simplifying to: $\frac{2.5 + 0}{\sqrt{5 + 0}} = \frac{2.5}{\sqrt{5}}.$
   • This gives the positive horizontal asymptote: $\frac{2.5}{\sqrt{5}} = \frac{2.5}{\sqrt{5}}$.

3. Horizontal asymptotes:

   • The horizontal asymptote as $x \to \infty$ is $\frac{2.5}{\sqrt{5}}$.
   • The same calculation can be applied as $x \to -\infty$, but there will be a negative sign for $x$, yielding the asymptote $-\frac{2.5}{\sqrt{5}}$.

Thus, the horizontal asymptotes are: $\boxed{\pm \frac{2.5}{\sqrt{5}}}.$

Would you like me to go over any specific steps in more detail or explain any other part of this?

Here are five follow-up questions you might find useful:

1. What is the importance of horizontal asymptotes in understanding the behavior of a function?
2. How do we generally approach the limit of functions involving square roots at infinity?
3. What other techniques can be used to find asymptotes of more complex functions?
4. Can you think of an example where a function has no horizontal asymptote?
5. How do horizontal asymptotes differ from vertical asymptotes?

Tip: Always remember to divide by the highest degree term when evaluating limits of rational functions at infinity to simplify the process.