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To solve this limit problem, let’s analyze the expression step-by-step:

\[
\lim_{{x \to -\infty}} \left( \frac{8}{x} - \frac{x}{9} \right)
\]

### Step 1: Separate the Terms
Since the limit of a sum (or difference) is the sum (or difference) of the limits, we can break it down as:

\[
\lim_{{x \to -\infty}} \frac{8}{x} - \lim_{{x \to -\infty}} \frac{x}{9}
\]

### Step 2: Evaluate Each Term Individually
1. **First Term**: \(\lim_{{x \to -\infty}} \frac{8}{x}\)

   As \(x \to -\infty\), \(\frac{8}{x}\) approaches 0 because \(8\) is a constant and \(\frac{1}{x}\) approaches 0 as \(x\) becomes very large in magnitude. Thus,

   \[
   \lim_{{x \to -\infty}} \frac{8}{x} = 0
   \]

2. **Second Term**: \(\lim_{{x \to -\infty}} \frac{x}{9}\)

   As \(x \to -\infty\), \(\frac{x}{9}\) also approaches \(-\infty\), because dividing by a positive constant (9) doesn’t change the sign of \(x\). Thus,

   \[
   \lim_{{x \to -\infty}} \frac{x}{9} = -\infty
   \]

### Step 3: Combine the Results
Now we combine the results from each term:

\[
0 - (-\infty) = -\infty
\]

### Conclusion
The limit diverges to \(-\infty\). Therefore, the answer is:

\[
\text{DNE (Does Not Exist)}
\]

Would you like more details or have any questions?

---

1. How do limits behave when \( x \) approaches positive infinity?
2. Why does dividing a constant by a variable approaching infinity result in zero?
3. How would the limit change if \( x \to \infty \) instead of \( x \to -\infty \)?
4. What are some cases when a limit does not exist?
5. How can we verify this result using graphical analysis?

**Tip:** When evaluating limits at infinity, break down complex expressions into simpler terms to analyze each part separately.

Find the limit, if it exists: lim(x→−∞) ((8/x) - (x/9))

Explore how to find the limit of the expression (8/x - x/9) as x approaches negative infinity. This calculus problem covers limits at infinity and the application of limit laws. Suitable for college-level calculus students.

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