https://r2.erweima.ai/mathbot/image/1726720479850m6zde5zo.jpg

We are tasked with finding the limit:

\[
\lim_{x \to \infty} \frac{3x^2 - 5x + 1}{9 + 8x - 2x^2}
\]

### Step 1: Focus on the dominant terms
For very large values of \(x\), the higher-degree terms dominate the behavior of the function. So, we focus on the terms with the highest degree of \(x^2\) in both the numerator and denominator.

- Numerator: \(3x^2 - 5x + 1\), where \(3x^2\) is dominant.
- Denominator: \(9 + 8x - 2x^2\), where \(-2x^2\) is dominant.

Thus, for large \(x\), the function behaves approximately like:

\[
\frac{3x^2}{-2x^2}
\]

### Step 2: Simplify the dominant term expression
Now, simplify this ratio:

\[
\frac{3x^2}{-2x^2} = \frac{3}{-2} = -\frac{3}{2}
\]

### Conclusion:
As \(x \to \infty\), the limit of the given function is \(-\frac{3}{2}\).

Thus, the correct answer is:

\[
\boxed{e) -\frac{3}{2}}
\]

Would you like a further breakdown of the process or have additional questions?

### Related Questions:
1. What is the limit if the denominator has a higher degree than the numerator?
2. How does the behavior of linear terms affect limits at infinity?
3. How can you apply L'Hopital's Rule to this type of limit problem?
4. What happens to the limit when the degrees of the numerator and denominator are equal?
5. How do constants in the polynomial affect the limit at infinity?

### Tip:
When calculating limits at infinity for rational functions, always compare the degrees of the highest-power terms in the numerator and denominator first.

Nilai dari \( \lim_{x \to \infty} \frac{3x^2 - 5x + 1}{9 + 8x - 2x^2} \) adalah...

Learn how to solve the limit \( \lim_{x \to \infty} \frac{3x^2 - 5x + 1}{9 + 8x - 2x^2} \) by focusing on the dominant terms of the rational function. This step-by-step guide covers key calculus concepts like limits at infinity and simplification of rational functions. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation