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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!

Find the limit: lim (x -> -∞) [5/x - x/4].

Learn how to calculate the limit of the expression [5/x - x/4] as x approaches negative infinity. This problem focuses on key calculus concepts such as limits and the behavior of functions at infinity. Suitable for students in Grades 11-12.

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