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.