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Let's solve the given problem step-by-step. The expression to evaluate is:

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

### Step 1: Analyzing the Integrand
The integrand is \( \ln\left(1 + \frac{1}{\sqrt{x}}\right) \). For large \( x \), \( \frac{1}{\sqrt{x}} \) becomes small. We can use the approximation:
\[
\ln\left(1 + \frac{1}{\sqrt{x}}\right) \approx \frac{1}{\sqrt{x}} \quad \text{for large } x.
\]

### Step 2: Simplifying the Integral
To get a better understanding, let's approximate the integral by replacing the logarithmic term with \( \frac{1}{\sqrt{x}} \):
\[
\int_1^n \ln\left(1 + \frac{1}{\sqrt{x}}\right) \, dx \approx \int_1^n \frac{1}{\sqrt{x}} \, dx.
\]

### Step 3: Evaluating the Approximate Integral
The integral of \( \frac{1}{\sqrt{x}} \) is straightforward:
\[
\int_1^n \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} \Big|_1^n = 2\sqrt{n} - 2.
\]

### Step 4: Limit Evaluation
Now, let's plug this approximation into the limit:
\[
\lim_{n \to \infty} \frac{1}{\sqrt{n}} \left( 2\sqrt{n} - 2 \right).
\]

Distribute \( \frac{1}{\sqrt{n}} \):
\[
\lim_{n \to \infty} \left( 2 - \frac{2}{\sqrt{n}} \right).
\]

As \( n \) approaches infinity, \( \frac{2}{\sqrt{n}} \) approaches 0, so the expression simplifies to:
\[
\lim_{n \to \infty} \left( 2 - 0 \right) = 2.
\]

Thus, the value of the given limit is:
\[
\boxed{2}.
\]

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

Here are 5 related questions for further exploration:
1. How can we rigorously justify the approximation \( \ln(1 + \frac{1}{\sqrt{x}}) \approx \frac{1}{\sqrt{x}} \) for large \( x \)?
2. What happens if the integrand is changed to \( \ln(1 + \frac{1}{x}) \) instead?
3. Can the Dominated Convergence Theorem be applied to this problem for a rigorous solution?
4. How would the result change if the upper limit of integration were something other than \( n \)?
5. What role does the factor \( \frac{1}{\sqrt{n}} \) play in the limit?

**Tip:** When evaluating limits involving integrals, approximating the integrand can often provide insights into the behavior of the entire expression.

Evaluate the limit: lim(n → ∞) (1 / √n) ∫[1 to n] ln(1 + (1/√x)) dx

Step-by-step solution of the limit lim(n → ∞) (1 / √n) ∫[1 to n] ln(1 + (1/√x)) dx, focusing on integral evaluation, logarithmic approximation, and limits. Suitable for university-level calculus students.

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