The given limit expression is:

$\lim_{x \to -\infty} \left( \frac{5}{x} - \frac{x}{4} \right)$

Let's evaluate the limit step by step.

1. Evaluate the first term $\frac{5}{x}$ as $x \to -\infty$:

• As $x \to -\infty$, the fraction $\frac{5}{x}$ approaches 0 because $5$ is a constant and the denominator $x$ becomes very large in magnitude. Therefore: $\lim_{x \to -\infty} \frac{5}{x} = 0$

2. Evaluate the second term $\frac{x}{4}$ as $x \to -\infty$:

• $\frac{x}{4}$ simplifies to a linear expression where $x \to -\infty$ means $\frac{x}{4} \to -\infty$. Thus: $\lim_{x \to -\infty} \frac{x}{4} = -\infty$

3. Combine the results:

Now we combine the two results: $\lim_{x \to -\infty} \left( \frac{5}{x} - \frac{x}{4} \right) = 0 - (-\infty) = \infty$

Thus, the correct limit is $\infty$, not $-\infty$.

Would you like further details or clarifications?

Here are 5 related questions to expand on this:

1. What happens to $\frac{1}{x}$ as $x \to \infty$ or $x \to -\infty$?
2. How do linear terms like $\frac{x}{4}$ behave as $x \to \infty$?
3. How do you apply L'Hôpital's Rule to evaluate limits of indeterminate forms?
4. What is the difference between limits approaching $+\infty$ and $-\infty$?
5. Can limits of rational functions at infinity sometimes result in finite values?

Tip: When evaluating limits at infinity, focus on the dominant terms (the ones that grow the fastest) to simplify your work!