https://file.aiquickdraw.com/mathbot/file/1726497818573y2zseg1f.jpg

The given problem is asking to find the limit:

\[
\lim_{x \to \infty} \frac{2x - 3}{x^2 - 3x + 2}
\]

Let's solve this step by step.

### Step 1: Analyze the degrees of the polynomial.
- The numerator is \(2x - 3\), a polynomial of degree 1.
- The denominator is \(x^2 - 3x + 2\), a polynomial of degree 2.

### Step 2: Divide both numerator and denominator by the highest power of \(x\), which is \(x^2\).
\[
\lim_{x \to \infty} \frac{\frac{2x - 3}{x^2}}{\frac{x^2 - 3x + 2}{x^2}} = \lim_{x \to \infty} \frac{\frac{2}{x} - \frac{3}{x^2}}{1 - \frac{3}{x} + \frac{2}{x^2}}
\]

### Step 3: Simplify the expression as \(x \to \infty\).
- \(\frac{2}{x} \to 0\)
- \(\frac{3}{x^2} \to 0\)
- \(\frac{3}{x} \to 0\)
- \(\frac{2}{x^2} \to 0\)

Thus, the limit simplifies to:
\[
\lim_{x \to \infty} \frac{0 - 0}{1 - 0 + 0} = 0
\]

### Conclusion:
The value of the limit is 0.

So, the correct answer is **C. 0**.

Would you like further details or explanations? 

Here are 5 related questions for practice:
1. What is the limit \(\lim_{x \to \infty} \frac{3x^2 + 2x}{x^2 - 4x + 1}\)?
2. How would the limit change if the degree of the numerator were higher than the denominator?
3. What is the general strategy for finding limits of rational functions as \(x \to \infty\)?
4. What happens to the limit when the degree of the denominator is higher than the numerator?
5. How do horizontal asymptotes relate to limits as \(x \to \infty\)?

**Tip:** When finding limits of rational functions, always compare the degrees of the numerator and denominator to predict the behavior as \(x\) approaches infinity.

Nilai lim x→∞ (2x-3) / (x^2 - 3x + 2) = ....

Find the limit of (2x - 3) / (x^2 - 3x + 2) as x approaches infinity with a step-by-step solution, covering key calculus concepts like polynomial division and limits. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation