https://file.aiquickdraw.com/mathbot/file/17291013740758g151whm.jpg

The given problem asks to find the limit:

\[
\lim_{x \to +\infty} \frac{x^2}{1 + x + 2x^2}
\]

### Solution:

1. **Analyze the highest degree of \(x\)**: The numerator is \(x^2\), and the denominator is \(1 + x + 2x^2\). Notice that both the numerator and the denominator contain terms of degree 2, which are the dominant terms as \(x \to +\infty\).

2. **Divide the numerator and denominator by \(x^2\)**: To simplify the expression, divide every term by \(x^2\), the highest power of \(x\) in the denominator.

\[
\frac{x^2}{1 + x + 2x^2} = \frac{\frac{x^2}{x^2}}{\frac{1}{x^2} + \frac{x}{x^2} + \frac{2x^2}{x^2}} = \frac{1}{\frac{1}{x^2} + \frac{1}{x} + 2}
\]

3. **Simplify as \(x \to +\infty\)**:
   - \(\frac{1}{x^2} \to 0\)
   - \(\frac{1}{x} \to 0\)

   So, the limit simplifies to:

\[
\lim_{x \to +\infty} \frac{1}{0 + 0 + 2} = \frac{1}{2}
\]

### Final Answer:
\[
\lim_{x \to +\infty} \frac{x^2}{1 + x + 2x^2} = \frac{1}{2}
\]

Would you like further details on any part of this explanation? Here are a few related questions to deepen your understanding:

1. What happens to lower-degree terms when \(x \to \infty\)?
2. How do we find the dominant term in both the numerator and denominator of a rational function?
3. Why is dividing by the highest power of \(x\) a useful strategy for finding limits at infinity?
4. What would the limit be if the denominator had a higher degree than the numerator?
5. Can you apply a similar approach to the limit \( \lim_{x \to +\infty} \frac{x^3}{1 + x^2 + 2x^3} \)?

**Tip**: Always identify the highest degree terms in polynomials when finding limits at infinity. These terms dictate the behavior of the function as \(x \to \infty\).

Find: lim x -> +∞ (x^2) / (1 + x + 2x^2)

Learn how to solve the limit problem lim x -> +∞ (x^2) / (1 + x + 2x^2) using rational function analysis. This problem covers key calculus concepts including identifying dominant terms and simplifying limits at infinity. Suitable for high school 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