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Given the expression:

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

we need to evaluate this limit. Let's break this down into simpler steps and examine the behavior of the integral as \( n \to \infty \).

### Step-by-Step Solution

1. **Understand the Integral**:
   - The integral \(\int_1^n \ln \left( 1 + \frac{1}{\sqrt{x}} \right) \, dx\) is challenging to compute directly. However, we can gain insight by examining the behavior of the integrand for large \(x\).

2. **Approximate the Logarithmic Function**:
   - For large \(x\), \(\frac{1}{\sqrt{x}}\) is small. Thus, we use the approximation \(\ln(1 + u) \approx u\) when \(u \to 0\). Here, \(u = \frac{1}{\sqrt{x}}\), so:
     \[
     \ln \left( 1 + \frac{1}{\sqrt{x}} \right) \approx \frac{1}{\sqrt{x}}.
     \]
   - The integrand then simplifies approximately to \(\frac{1}{\sqrt{x}}\) for large \(x\).

3. **Approximate the Integral**:
   - Using the approximation, the integral becomes:
     \[
     \int_1^n \ln \left( 1 + \frac{1}{\sqrt{x}} \right) \, dx \approx \int_1^n \frac{1}{\sqrt{x}} \, dx.
     \]
   - Compute this integral:
     \[
     \int_1^n \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} \bigg|_1^n = 2\sqrt{n} - 2.
     \]

4. **Plug into the Limit**:
   - Now, substitute this result into the original limit:
     \[
     \lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( 2\sqrt{n} - 2 \right).
     \]
   - Simplify the expression:
     \[
     \lim_{n \to \infty} \left( 2 - \frac{2}{\sqrt{n}} \right).
     \]

5. **Evaluate the Limit**:
   - As \(n \to \infty\), \(\frac{2}{\sqrt{n}} \to 0\). Therefore, the limit simplifies to:
     \[
     \lim_{n \to \infty} \left( 2 - 0 \right) = 2.
     \]

### Final Answer

\[
\lim_{n \to \infty} \frac{1}{\sqrt{n}} \int_1^n \ln \left( 1 + \frac{1}{\sqrt{x}} \right) \, dx = 2.
\]

Would you like a more detailed explanation or have any questions about this process?

### Related Questions

1. How do you approximate integrals using asymptotic analysis?
2. What is the small \(u\) approximation of \(\ln(1 + u)\) and when is it valid?
3. How can we use substitutions to simplify integrals involving logarithmic functions?
4. Why does the dominant behavior of the integral determine the limit in problems like this?
5. How does changing the limits of integration affect the final answer of a limit problem?

### Tip
When evaluating limits involving integrals, it often helps to approximate the integrand for large values of the variable to simplify calculations.

数列极限 lim_{n->∞} (1/sqrt(n)) ∫_1^n ln(1 + 1/sqrt(x)) dx = ?

This problem explores the limit of an integral involving logarithmic functions and square roots as n approaches infinity. Learn the step-by-step approximation technique to evaluate the integral and find the final limit.

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