We need to evaluate the limit:

$\lim_{x \to \infty} \frac{9x^6 - x}{x^3 + 8}$

Step 1: Identify the Highest Power of $x$

The highest power in the numerator is $x^6$ and in the denominator is $x^3$.

Step 2: Divide by the Highest Power in the Denominator

Since the highest power in the denominator is $x^3$, we divide every term by $x^3$:

$\frac{9x^6 - x}{x^3 + 8} = \frac{9x^6/x^3 - x/x^3}{(x^3/x^3) + (8/x^3)}$

which simplifies to:

$\frac{9x^3 - \frac{1}{x^2}}{1 + \frac{8}{x^3}}.$

Step 3: Evaluate the Limit

As $x \to \infty$:

• $\frac{1}{x^2} \to 0$,
• $\frac{8}{x^3} \to 0$,
• $9x^3$ remains dominant.

Thus, the fraction simplifies to:

$\frac{9x^3}{1} = 9x^3.$

Since $9x^3 \to \infty$, the limit does not exist (DNE) because it diverges to infinity.

Final Answer:

$\text{DNE (Does Not Exist)}$

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Would you like a breakdown of a similar problem? Here are five related questions:

1. Find $\lim_{x \to \infty} \frac{5x^4 - 2x}{x^4 + 3}$.
2. Evaluate $\lim_{x \to \infty} \frac{x^5 - 7}{2x^5 + 10}$.
3. Determine whether $\lim_{x \to \infty} \frac{3x^3 - x}{x^5 + 1}$ exists.
4. Compute $\lim_{x \to \infty} \frac{4x^2 + 10x}{2x^2 - 5x}$.
5. Find $\lim_{x \to \infty} \frac{x^7 + x^6}{x^7 - x^5}$.

Tip:

For rational functions, compare the degrees of the numerator and denominator. If the numerator's degree is higher, the function tends to infinity. If the denominator's degree is higher, the function tends to zero.