The given limit expression is:

$\lim_{x \to \infty} \left( \frac{x + 5}{x} \sqrt{x} \right)$

Step-by-Step Solution:

1. Simplify the expression inside the limit: $\frac{x + 5}{x} = \frac{x}{x} + \frac{5}{x} = 1 + \frac{5}{x}$ So, the original limit becomes: $\lim_{x \to \infty} \left( \left( 1 + \frac{5}{x} \right) \sqrt{x} \right)$

2. Take the limit as $x \to \infty$: As $x \to \infty$, the term $\frac{5}{x} \to 0$. Hence, the expression simplifies to: $\left( 1 + 0 \right) \sqrt{x} = \sqrt{x}$

3. Final Limit: The limit of $\sqrt{x}$ as $x \to \infty$ is infinite. Therefore: $\lim_{x \to \infty} \left( \frac{x+5}{x} \sqrt{x} \right) = \infty$

Thus, the limit is:

$\boxed{\infty}$

Would you like more details or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. How do we handle limits with square roots in the denominator?
2. What happens if we have a negative power in the limit as $x \to \infty$?
3. How would the result change if the expression inside the square root was different?
4. What is the behavior of limits that involve logarithmic functions as $x \to \infty$?
5. Can you explain how L'Hopital's Rule could apply to this type of limit?

Tip: Always simplify the expression as much as possible before applying the limit to identify terms that vanish or dominate.