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Let's solve the given limit:

\[
\lim_{{x \to \infty}} \frac{2x^5 + 9x^4 - 4x^3}{26 - 8x^5}
\]

### Step 1: Factor out the highest power of \(x\) in the numerator and the denominator.
The highest power of \(x\) in both the numerator and denominator is \(x^5\). Factor \(x^5\) out:

\[
\lim_{{x \to \infty}} \frac{x^5\left(2 + \frac{9}{x} - \frac{4}{x^2}\right)}{x^5\left(\frac{26}{x^5} - 8\right)}
\]

### Step 2: Simplify the expression.
Since \(x^5\) is common in both the numerator and the denominator, it cancels out:

\[
\lim_{{x \to \infty}} \frac{2 + \frac{9}{x} - \frac{4}{x^2}}{\frac{26}{x^5} - 8}
\]

### Step 3: Apply the limit.
As \(x \to \infty\), the terms with \( \frac{9}{x} \), \( \frac{4}{x^2} \), and \( \frac{26}{x^5} \) approach zero:

\[
\lim_{{x \to \infty}} \frac{2 + 0 - 0}{0 - 8} = \frac{2}{-8} = -\frac{1}{4}
\]

### Final Answer:
\[
\lim_{{x \to \infty}} \frac{2x^5 + 9x^4 - 4x^3}{26 - 8x^5} = -\frac{1}{4}
\]

Would you like more details or have any questions? 

Here are five related questions you might find interesting:
1. How does the degree of the polynomial in the numerator and denominator affect the limit as \(x\) approaches infinity?
2. What happens if the highest power of \(x\) in the denominator is higher than in the numerator?
3. Can limits involving infinity always be evaluated by factoring out the highest power of \(x\)?
4. How would the solution change if the sign of the highest power of \(x\) in the denominator were positive?
5. How can L'Hôpital's Rule be applied to similar limit problems?

**Tip:** When dealing with limits approaching infinity, always factor out the highest power of \(x\) to simplify the expression and make the limit easier to evaluate.

Use the properties of limits to find the given limit: lim (x→∞) (2x^5 + 9x^4 - 4x^3) / (26 - 8x^5)

Learn how to find the limit of a rational function as x approaches infinity by factoring out the highest power of x. This step-by-step solution explains key concepts in calculus, including limits and polynomial functions. Suitable for students in Grades 11-12.

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