Write the precise definition and limit of lim approaching -infinity (9/x^3)=0

The limit of the function \( \frac{9}{x^3} \) as \( x \) approaches negative infinity is defined as:

\[
\lim_{x \to -\infty} \frac{9}{x^3}
\]

### Step-by-step process:

1. **Analyze the behavior of \( x^3 \) as \( x \to -\infty \):**
   - As \( x \to -\infty \), \( x^3 \) becomes very large in magnitude but negative, meaning \( x^3 \to -\infty \).
   
2. **Effect on \( \frac{9}{x^3} \):**
   - Since \( x^3 \to -\infty \), the expression \( \frac{9}{x^3} \) approaches \( 0 \) but remains negative because the denominator is negative for negative \( x \).
   
3. **Conclusion:**
   - The value of \( \frac{9}{x^3} \) approaches 0 from the negative side as \( x \to -\infty \).
   - Hence, the limit is 0.

\[
\lim_{x \to -\infty} \frac{9}{x^3} = 0
\]

This result tells us that as \( x \) becomes very large in the negative direction, the function \( \frac{9}{x^3} \) tends toward 0.

Would you like more details or further clarification on this? Here are some related questions you could explore:

1. What is the limit of \( \frac{9}{x^2} \) as \( x \to -\infty \)?
2. How does the behavior of even powers of \( x \) differ from odd powers as \( x \to -\infty \)?
3. What is the limit of \( \frac{-9}{x^3} \) as \( x \to -\infty \)?
4. How would the limit change if \( x \to \infty \) instead of \( x \to -\infty \)?
5. Can we use L'Hôpital's Rule to evaluate limits involving \( \frac{9}{x^3} \)?

**Tip**: As \( x \to \infty \), rational functions with higher powers of \( x \) in the denominator generally tend toward 0, regardless of the numerator.

Explore how to evaluate the limit of the function 9/x^3 as x approaches negative infinity. This step-by-step solution covers the behavior of rational functions and their limits at infinity, focusing on the properties of negative powers of x. Ideal for students in Grades 11-12 studying calculus.

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