To solve the given limit problem:

$\lim_{x \to +\infty} \frac{(4x^2 + x + 1)(\sqrt{x} + 2)}{(3x\sqrt{x} - 1)(\sqrt{x} + 1)^2}$

Step-by-Step Solution:

1. Identify the highest degree terms:

   • In the numerator: The term $4x^2$ dominates in $(4x^2 + x + 1)$ as $x \to \infty$.
   • In the denominator:
     • In $3x\sqrt{x} - 1$, the term $3x\sqrt{x}$ dominates.
     • In $(\sqrt{x} + 1)^2$, expand: $(\sqrt{x} + 1)^2 = x + 2\sqrt{x} + 1$, so $x$ dominates.

2. Simplify the numerator: $(4x^2 + x + 1)(\sqrt{x} + 2) \sim (4x^2)(\sqrt{x}) = 4x^{2.5}, \quad \text{as } x \to \infty.$

3. Simplify the denominator: $(3x\sqrt{x} - 1)(\sqrt{x} + 1)^2 \sim (3x\sqrt{x})(x) = 3x^{2.5}, \quad \text{as } x \to \infty.$

4. Combine the results: $\frac{(4x^2 + x + 1)(\sqrt{x} + 2)}{(3x\sqrt{x} - 1)(\sqrt{x} + 1)^2} \sim \frac{4x^{2.5}}{3x^{2.5}} = \frac{4}{3}.$

Final Answer:

$\lim_{x \to +\infty} \frac{(4x^2 + x + 1)(\sqrt{x} + 2)}{(3x\sqrt{x} - 1)(\sqrt{x} + 1)^2} = \frac{4}{3}.$

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5 Related Questions:

1. What is the general method for finding the dominant term in a rational function as $x \to \infty$?
2. How does the behavior of square root terms affect limits involving infinity?
3. Can you explain how to expand and simplify polynomials involving square roots?
4. What are some typical applications of limits at infinity in calculus?
5. How do you apply L'Hopital's Rule for limits approaching infinity, and is it needed here?

Tip:

When solving limits as $x \to \infty$, focus on the highest degree terms in both the numerator and the denominator