https://file.aiquickdraw.com/mathbot/file/1726624124205qfaj37lt.jpg

The given mathematical expression is asking for the limit of a rational function as \( x \) approaches infinity:

\[
\lim_{x \to \infty} \frac{-7x^2 + 3x - 6}{-2x + 1}
\]

### Steps to solve:

1. **Identify the degrees of the polynomials:**
   - The numerator is \( -7x^2 + 3x - 6 \), which is a quadratic polynomial (degree 2).
   - The denominator is \( -2x + 1 \), which is a linear polynomial (degree 1).

2. **Since the degree of the numerator (2) is greater than the degree of the denominator (1)**, the behavior of the function as \( x \) approaches infinity will be dominated by the highest degree terms in both the numerator and the denominator. Specifically, it simplifies to:

\[
\frac{-7x^2}{-2x}
\]

3. **Simplify the expression**:

\[
\frac{-7x^2}{-2x} = \frac{7x}{2}
\]

As \( x \to \infty \), \( \frac{7x}{2} \) approaches infinity.

### Conclusion:
\[
\lim_{x \to \infty} \frac{-7x^2 + 3x - 6}{-2x + 1} = \infty
\]

This means the value of the limit is infinite.

Let me know if you'd like more details or further clarifications!

### Related questions:
1. What happens if the degrees of the numerator and denominator are the same?
2. How would you handle limits if \( x \to -\infty \)?
3. What if both the numerator and denominator were linear polynomials?
4. Can you compute limits approaching a specific finite number?
5. How would the limit change if the highest degree terms in both the numerator and denominator had different signs?

### Tip:
When dealing with rational functions, always focus on the terms with the highest degrees as \( x \to \infty \), since they dominate the behavior.

Tentukan nilai dari \( \lim_{x \to \infty} \frac{-7x^2 + 3x - 6}{-2x + 1} \)

Learn how to calculate the limit of the rational function \( \frac{-7x^2 + 3x - 6}{-2x + 1} \) as \( x \) approaches infinity. This problem explores key calculus concepts, including the asymptotic behavior of polynomials and rational functions. Suitable for high school students in 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