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We are tasked with solving the limit:

\[
\lim_{x \to 0} \left( \frac{1}{x} \left( \sqrt{\frac{1}{x^2} + 1} - \frac{1}{x} \right) + x \ln\left(1 + a^{1/x}\right) \right),
\]
where \( a > 0 \) and \( a \neq 1 \).

### Step 1: Analyze the First Term
The first term is:
\[
\frac{1}{x} \left( \sqrt{\frac{1}{x^2} + 1} - \frac{1}{x} \right).
\]
Simplifying the expression inside the square root:
\[
\sqrt{\frac{1}{x^2} + 1} = \frac{\sqrt{1 + x^2}}{|x|}.
\]
For \( x \to 0 \), we approximate:
\[
\sqrt{1 + x^2} \approx 1 + \frac{x^2}{2}.
\]
Thus:
\[
\frac{1}{x} \left( \sqrt{\frac{1}{x^2} + 1} - \frac{1}{x} \right) \approx \frac{1}{x} \left( \frac{1 + \frac{x^2}{2}}{x} - \frac{1}{x} \right).
\]
This simplifies to:
\[
\frac{1}{x} \left( \frac{1}{x} + \frac{x}{2} - \frac{1}{x} \right) = \frac{x}{2}.
\]
As \( x \to 0 \), this term tends to 0.

### Step 2: Analyze the Second Term
The second term is:
\[
x \ln\left(1 + a^{1/x}\right).
\]
As \( x \to 0 \), we can expand this term using logarithmic approximation. Since \( a^{1/x} \) grows rapidly for small \( x \), we know:
\[
\ln\left(1 + a^{1/x}\right) \sim \ln\left(a^{1/x}\right) = \frac{1}{x} \ln a.
\]
Thus:
\[
x \ln\left(1 + a^{1/x}\right) \approx x \cdot \frac{1}{x} \ln a = \ln a.
\]

### Step 3: Combine the Results
The first term tends to 0, and the second term tends to \( \ln a \). Therefore, the overall limit becomes:
\[
\lim_{x \to 0} \left( 0 + \ln a \right) = \ln a.
\]

For the given choices:
- \( \ln a = 1 \) only when \( a = e \), but \( a \) was not specified as \( e \), and the options do not suggest this.

Thus, the correct answer seems to be **(D) None of these**.

Would you like any additional clarification or details?

### Follow-up questions:
1. What is the general approach to evaluating limits involving logarithmic terms?
2. How do approximations work in solving limits involving square roots?
3. Can you explain the approximation of \( \ln(1 + a^{1/x}) \) for small \( x \)?
4. What are other common methods to simplify expressions with indeterminate forms?
5. How does the behavior of \( a^{1/x} \) as \( x \to 0 \) affect the result?

### Tip:
When evaluating limits, using Taylor series expansions or approximations around small values of \( x \) can help simplify complicated expressions.

Find the limit of \( \lim_{x \to 0} \left( \frac{1}{x} \left( \sqrt{\frac{1}{x^2} + 1} - \frac{1}{x} \right) + x \ln\left(1 + a^{1/x}\right) \right), \ a > 0, a \neq 1 \).

This page provides a step-by-step solution to the limit \( \lim_{x \to 0} \left( \frac{1}{x} \left( \sqrt{\frac{1}{x^2} + 1} - \frac{1}{x} \right) + x \ln\left(1 + a^{1/x}\right) \right) \). The solution explores logarithmic approximations and square root simplifications for small values of x. Ideal for advanced calculus students.

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