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The given limit is:

\[
\lim_{x \to \infty} \frac{2x\sqrt{x} - x + 3}{\frac{3}{x^{\frac{1}{2}}}}
\]

To evaluate this limit, let's break it down step by step:

### Step 1: Simplify the denominator
The denominator is \(\frac{3}{x^{\frac{1}{2}}} = 3x^{-\frac{1}{2}}\). So, the expression becomes:

\[
\frac{2x\sqrt{x} - x + 3}{3x^{-\frac{1}{2}}}
\]

### Step 2: Multiply numerator and denominator by \(x^{\frac{1}{2}}\)
We multiply both the numerator and the denominator by \(x^{\frac{1}{2}}\) to eliminate the negative exponent in the denominator:

\[
\frac{x^{\frac{1}{2}}(2x\sqrt{x} - x + 3)}{3}
\]

Simplifying the terms in the numerator:
- \(x^{\frac{1}{2}} \cdot 2x\sqrt{x} = 2x^{\frac{3}{2}} \cdot x^{\frac{1}{2}} = 2x^2\)
- \(x^{\frac{1}{2}} \cdot (-x) = -x^{\frac{3}{2}}\)
- \(x^{\frac{1}{2}} \cdot 3 = 3x^{\frac{1}{2}}\)

So the expression becomes:

\[
\frac{2x^2 - x^{\frac{3}{2}} + 3x^{\frac{1}{2}}}{3}
\]

### Step 3: Analyze the limit as \(x \to \infty\)
As \(x\) approaches infinity, we observe the terms in the numerator:
- The term \(2x^2\) dominates because it has the highest power of \(x\).
- The other terms (\(-x^{\frac{3}{2}}\) and \(3x^{\frac{1}{2}}\)) become negligible as \(x \to \infty\).

Thus, the expression approaches:

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

### Conclusion:
The limit is infinite.

\[
\boxed{\infty}
\]

Would you like further clarification on any step, or to explore related problems?

Here are 5 related questions:
1. What happens if the power of \(x\) in the denominator is higher than in the numerator?
2. How do limits behave when approaching \(0\) instead of infinity?
3. Can L'Hopital's rule be used here? Why or why not?
4. How does multiplying by \(x^{\frac{1}{2}}\) help simplify the problem?
5. What techniques can be applied to solve indeterminate limits?

**Tip:** When evaluating limits at infinity, always focus on the highest power terms in both the numerator and denominator to simplify the expression.

Evaluate the limit: lim (x -> infinity) [(2x√x - x + 3) / (3 / x^(1/2))]

Learn how to evaluate the limit of the expression (2x√x - x + 3) / (3 / x^(1/2)) as x approaches infinity. This step-by-step solution breaks down key concepts in limits, simplification, and asymptotic behavior, suitable for high school and undergraduate calculus students.

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