Can you show me the steps to this problem: lim of x approaching infinity sqrt of 9x^6-x divided by x^3 +1

Certainly! Let's solve the limit of the function as \( x \) approaches infinity for the given expression:

\[
\lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1}
\]

### Step-by-Step Solution:

1. **Identify the highest powers of \( x \) in the numerator and denominator**:
   - In the numerator \( \sqrt{9x^6 - x} \), the highest power of \( x \) inside the square root is \( x^6 \).
   - In the denominator \( x^3 + 1 \), the highest power of \( x \) is \( x^3 \).

2. **Simplify the square root in the numerator**:
   \[
   \sqrt{9x^6 - x} = \sqrt{x^6(9 - \frac{1}{x^5})}
   \]
   Using the fact that \( \lim_{x \to \infty} \frac{1}{x^5} = 0 \), we approximate:
   \[
   \sqrt{9x^6 - x} \approx \sqrt{9x^6} = 3x^3
   \]

3. **Rewrite the original expression**:
   Now, the expression becomes approximately:
   \[
   \frac{\sqrt{9x^6 - x}}{x^3 + 1} \approx \frac{3x^3}{x^3 + 1}
   \]

4. **Divide both the numerator and denominator by \( x^3 \)**:
   \[
   \frac{3x^3}{x^3 + 1} = \frac{3}{1 + \frac{1}{x^3}}
   \]

5. **Evaluate the limit**:
   As \( x \to \infty \), \( \frac{1}{x^3} \to 0 \). Therefore, the expression simplifies to:
   \[
   \frac{3}{1 + 0} = 3
   \]

### Final Answer:
\[
\lim_{x \to \infty} \frac{\sqrt{9x^6 - x}}{x^3 + 1} = 3
\]

Let me know if you need more details, or have any questions!

Here are 5 related questions you might find interesting:
1. What happens if the highest power of \( x \) in the numerator is lower than the denominator?
2. How would this limit change if the coefficient of \( x^6 \) in the numerator was different?
3. What is the behavior of the function as \( x \to -\infty \)?
4. Can you apply L'Hopital's rule to solve this limit? Would it give the same result?
5. How do higher-order terms affect limits at infinity?

**Tip**: When dealing with limits as \( x \) approaches infinity, factor out the highest power of \( x \) to simplify the expression.

Solve the limit problem involving sqrt(9x^6 - x) divided by (x^3 + 1) as x approaches infinity. This detailed step-by-step solution covers the use of limits, simplification of radicals, and asymptotic behavior. Suitable for high school seniors or college-level students.

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