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We are given the limit:

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

### Step 1: Simplify the denominator
First, notice that the denominator can be rewritten as:

\[
\frac{3}{x^2} = 3x^{-2}
\]

So the expression becomes:

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

### Step 2: Multiply numerator and denominator by \( x^2 \)
To simplify further, we multiply both the numerator and the denominator by \( x^2 \), which eliminates the negative exponent in the denominator:

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

Now, distribute \( x^2 \) in the numerator:

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

### Step 3: Focus on the highest powers of \(x\)
Since we are taking the limit as \(x\) approaches infinity, the terms with the highest powers of \(x\) dominate the behavior of the expression. The term \(2x^3 \sqrt{x}\) (which is actually \(2x^{7/2}\)) will grow faster than the other terms as \(x\) goes to infinity. Therefore, we focus on that leading term.

### Step 4: Approximate the limit
The dominant term in the numerator is \(2x^{7/2}\). So, we approximate the limit as:

\[
\lim_{{x \to \infty}} \frac{2x^{7/2}}{3} = \infty
\]

### Conclusion
The limit diverges to infinity:

\[
\boxed{\infty}
\]

---

Would you like further details on any of the steps or concepts involved? Here are some related questions for practice:

1. What is the behavior of the function \(x \sqrt{x}\) as \(x\) approaches infinity?
2. How do you determine the dominant term when simplifying limits at infinity?
3. How does multiplying by powers of \(x\) affect the limit of a rational function?
4. Can you explain why lower-order terms can be ignored when finding limits at infinity?
5. How would you approach a limit problem where both numerator and denominator grow without bound?

**Tip:** When dealing with limits approaching infinity, focus on the highest-order term to determine the function's growth.

What is the limit of (2x√x - x + 3) / (3/x^2) as x approaches infinity?

Explore the step-by-step solution to finding the limit of the function (2x√x - x + 3) / (3/x^2) as x approaches infinity. This solution covers key concepts such as asymptotic behavior and dominance of higher-order terms, making it ideal for university-level students.

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