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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.

Find the limit of ((x + 5)/x) * sqrt(x) as x approaches infinity with step-by-step simplification. Learn key concepts such as simplifying fractions inside limits, square root behavior at infinity, and proper limit evaluation.

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