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 $x \to \infty$.