Let's evaluate the limit:

\[
\lim_{x \to \infty} \left( 2x^2 - 9x^2 - 3x \right)
\]

First, simplify the expression inside the limit:

\[
2x^2 - 9x^2 - 3x = -7x^2 - 3x
\]

Now we have:

\[
\lim_{x \to \infty} \left( -7x^2 - 3x \right)
\]

As \( x \to \infty \), \( x^2 \) dominates \( x \). Therefore, the \( -7x^2 \) term will grow without bound, and since it has a negative coefficient, the expression tends to \( -\infty \).

Thus, the limit is:

\[
\boxed{-\infty}
\]

Would you like more details or have any questions?

Here are some related questions:
1. What happens if the highest degree term is positive?
2. How does the degree of a polynomial affect the limit as \( x \to \infty \)?
3. What is the behavior of rational functions as \( x \to \infty \)?
4. Can a limit as \( x \to \infty \) result in a finite number?
5. How do exponential functions behave as \( x \to \infty \)?

**Tip:** Always look for the highest-degree term when evaluating limits at infinity for polynomials; it usually dominates the behavior of the function.

Explore how to evaluate the limit of the polynomial expression lim𝑥→∞(2𝑥^2−9𝑥^2−3𝑥). Learn how the highest degree term dominates the behavior of the function as x approaches infinity, resulting in a limit of negative infinity. Suitable for advanced high school or early college calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation