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The given mathematical expression involves taking the limit as \( n \to \infty \) of an integral expression. Specifically, it is:

\[
\lim_{n \to \infty} \frac{1}{\sqrt{n}} \int_1^n \ln \left( 1 + \frac{1}{\sqrt{x}} \right) dx
\]

### Step-by-Step Analysis:

1. **Understanding the structure**:
   - The expression is of the form \( \frac{1}{\sqrt{n}} \int_1^n f(x) dx \), where \( f(x) = \ln \left( 1 + \frac{1}{\sqrt{x}} \right) \).
   - We need to evaluate the limit of this integral as \( n \) approaches infinity.

2. **Behavior of the integrand**:
   - As \( x \to \infty \), \( \frac{1}{\sqrt{x}} \to 0 \), so the logarithmic term behaves as \( \ln(1) = 0 \) at large values of \( x \).
   - As \( x \to 1 \), the term becomes \( \ln \left( 1 + 1 \right) = \ln(2) \).

3. **Approximation of the integral for large \( n \)**:
   - For large \( x \), the integrand behaves like \( \ln \left( 1 + \frac{1}{\sqrt{x}} \right) \approx \frac{1}{\sqrt{x}} \) because for small values of \( y \), \( \ln(1 + y) \approx y \).

4. **Evaluate the limit**:
   The integral \( \int_1^n \ln \left( 1 + \frac{1}{\sqrt{x}} \right) dx \) grows more slowly than \( \sqrt{n} \) as \( n \to \infty \), and the behavior of the fraction \( \frac{1}{\sqrt{n}} \) in front will eventually dominate. Thus, the whole expression converges to a finite limit.

5. **Conclusion**:
   After evaluating the limit, the result simplifies to:

\[
\boxed{2}
\]

Would you like further details on the steps or clarifications?

### Related Questions:
1. How do you approximate integrals involving logarithmic functions?
2. What is the behavior of the natural logarithm function for large values of the argument?
3. How does the method of asymptotic analysis help in solving limits involving integrals?
4. What role does the \( 1/\sqrt{n} \) term play in this limit?
5. How can you simplify the logarithmic term for small values of \( 1/\sqrt{x} \)?

**Tip**: When dealing with limits of integrals, always analyze the behavior of the integrand as the variable tends to the boundaries of integration.

The mathematical expression involving the limit of an integral as n approaches infinity: 
\lim_{n \to \infty} \frac{1}{\sqrt{n}} \int_1^n \ln \left( 1 + \frac{1}{\sqrt{x}} \right) dx.

Analyze the limit of an integral expression involving logarithmic functions and asymptotic behavior as n approaches infinity. This problem uses key calculus concepts and is suitable for advanced undergraduate and graduate students.

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