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To evaluate the limit \(\lim_{{x \to \infty}} \frac{(10 - x)(6 + 10x)}{(3 - 10x)(8 + 9x)}\), we will analyze the behavior of the numerator and denominator as \(x\) approaches infinity.

### Step-by-step Solution:

1. **Expand the Numerator and Denominator:**
   - Numerator: \((10 - x)(6 + 10x) = 10(6 + 10x) - x(6 + 10x)\)
     \[
     = 60 + 100x - 6x - 10x^2 = -10x^2 + 94x + 60
     \]
   - Denominator: \((3 - 10x)(8 + 9x) = 3(8 + 9x) - 10x(8 + 9x)\)
     \[
     = 24 + 27x - 80x - 90x^2 = -90x^2 - 53x + 24
     \]

2. **Identify the Leading Terms:**
   - The leading term in the numerator is \(-10x^2\).
   - The leading term in the denominator is \(-90x^2\).

3. **Simplify the Limit Using the Leading Terms:**
   Since both the numerator and the denominator have the same degree (degree 2), we take the ratio of their leading coefficients:
   \[
   \lim_{{x \to \infty}} \frac{-10x^2}{-90x^2} = \frac{-10}{-90} = \frac{1}{9}
   \]

Therefore, the evaluated limit is:

\[
\lim_{{x \to \infty}} \frac{(10 - x)(6 + 10x)}{(3 - 10x)(8 + 9x)} = \frac{1}{9}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find the limits of functions with different degrees in the numerator and denominator?
2. What is the rule for determining the limit if the degrees of the numerator and denominator are not the same?
3. How does L'Hospital's Rule apply to limits involving infinity?
4. Can you explain the behavior of rational functions as \( x \to \infty \)?
5. How do asymptotes relate to limits at infinity?

### Tip:
For limits at infinity, focus on the highest degree terms in both the numerator and the denominator, as these dominate the behavior of the function as \(x\) becomes very large.

Evaluate the limit lim(x→∞) [(10 - x)(6 + 10x)] / [(3 - 10x)(8 + 9x)]

This guide explains how to evaluate the limit lim(x→∞) [(10 - x)(6 + 10x)] / [(3 - 10x)(8 + 9x)], focusing on the leading terms of the numerator and denominator. Learn the method of simplifying rational functions and analyzing their behavior as x approaches infinity. Suitable for Grades 11-12 and early college students.

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