https://file.aiquickdraw.com/mathbot/file/1729429707156dswlo68w.jpg

The given limit expression is:

\[
\lim_{x \to \infty} \frac{2x\sqrt{x} - x + 3}{\frac{3}{x^{\frac{2}{3}}}}
\]

Let's solve it step by step.

### Step 1: Simplify the denominator

The denominator is \(\frac{3}{x^{\frac{2}{3}}}\), which can be rewritten as:
\[
\frac{1}{x^{\frac{2}{3}}} = 3x^{-\frac{2}{3}}
\]
So the entire expression becomes:
\[
\lim_{x \to \infty} \frac{(2x\sqrt{x} - x + 3)}{3x^{-\frac{2}{3}}}
\]

### Step 2: Combine terms

We multiply both the numerator and denominator by \(x^{\frac{2}{3}}\) to eliminate the negative exponent in the denominator:

\[
= \lim_{x \to \infty} \frac{x^{\frac{2}{3}}(2x\sqrt{x} - x + 3)}{3}
\]

Now simplify each term in the numerator:

- \(2x\sqrt{x} = 2x^{\frac{3}{2}}\)
- The \(x\) term becomes \(x^{\frac{2}{3}} \cdot x = x^{\frac{5}{3}}\)
- The constant \(3\) becomes \(3x^{\frac{2}{3}}\)

Thus, the expression becomes:

\[
= \lim_{x \to \infty} \frac{2x^{\frac{5}{3}} - x^{\frac{5}{3}} + 3x^{\frac{2}{3}}}{3}
\]

### Step 3: Leading term analysis

As \(x \to \infty\), the highest power term in the numerator dominates. Here, the \(x^{\frac{5}{3}}\) terms dominate the growth. So, we focus on those terms:

\[
= \lim_{x \to \infty} \frac{(2 - 1)x^{\frac{5}{3}}}{3} = \lim_{x \to \infty} \frac{x^{\frac{5}{3}}}{3}
\]

As \(x \to \infty\), this expression tends to infinity.

### Final Answer:

\[
\lim_{x \to \infty} \frac{2x\sqrt{x} - x + 3}{\frac{3}{x^{\frac{2}{3}}}} = \infty
\]

The limit diverges to infinity.

---

Would you like further details or clarifications on any of the steps?

Here are some additional related questions for you to consider:
1. How would the result change if the leading term had a different power of \(x\)?
2. What happens to the limit if the denominator were \(\frac{1}{x^2}\)?
3. How do you apply L'Hopital's Rule for limits at infinity with indeterminate forms?
4. Can we use dominant term analysis for limits at infinity in general? How?
5. What are the conditions for a limit to approach a finite number instead of infinity?

**Tip:** When working with limits at infinity, always identify the dominant terms with the highest powers of \(x\), as they will drive the behavior of the function as \(x \to \infty\).

lim x→∞ (2x√x - x + 3) / (3 / x^(2/3))

This limit problem focuses on dominant term analysis and asymptotic behavior. Follow the detailed solution to determine the limit of (2x√x - x + 3) / (3 / x^(2/3)) as x approaches infinity. Suitable for students studying Calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation