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To solve the given problem, we need to analyze the expression \( \ln(x + \sqrt{1 + x^2}) \) as \( x \to 0 \).

First, let's rewrite the expression inside the natural logarithm:

\[ \ln\left(x + \sqrt{1 + x^2}\right) \]

When \( x \to 0 \), we can use a Taylor expansion or approximation to simplify the expression inside the logarithm.

1. **Simplifying the square root term:**

\[ \sqrt{1 + x^2} \]

For small \( x \), we can use the approximation:

\[ \sqrt{1 + x^2} \approx 1 + \frac{x^2}{2} \]

2. **Substituting this approximation back into the expression:**

\[ x + \sqrt{1 + x^2} \approx x + 1 + \frac{x^2}{2} \]

Since \( x \) is small, \( x^2 \) is even smaller, so the dominant terms are:

\[ x + 1 \]

3. **Taking the natural logarithm:**

\[ \ln\left(x + \sqrt{1 + x^2}\right) \approx \ln(1 + x) \]

For \( x \) close to 0, we can use the Taylor expansion of \( \ln(1 + x) \):

\[ \ln(1 + x) \approx x \]

Thus,

\[ \ln\left(x + \sqrt{1 + x^2}\right) \approx x \]

So, as \( x \to 0 \),

\[ \ln\left(x + \sqrt{1 + x^2}\right) \sim x \]

This completes the solution to the problem.

Would you like any further details or have any questions? Here are some related questions you might consider:

1. What is the Taylor series expansion for the natural logarithm function \( \ln(1 + x) \)?
2. How do you approximate expressions using Taylor series?
3. What are other methods for approximating limits of functions as \( x \to 0 \)?
4. How does the approximation \( \sqrt{1 + x^2} \approx 1 + \frac{x^2}{2} \) derive?
5. How can we apply these techniques to more complex functions?

**Tip:** When approximating functions near a specific point, Taylor series expansion is a powerful tool to simplify and understand the behavior of the function.

Learn how to analyze the expression ln(x + sqrt(1 + x^2)) as x approaches 0 using the Taylor series expansion. This problem involves the natural logarithm and its approximation for small values of x. Suitable for advanced mathematics students.

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