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The limit you provided is:

\[
\lim_{x \to 0} \left( \frac{1}{x} \ln \left(1 + x\right) \right)
\]

Let's solve this limit step by step.

### Step 1: Determine the Indeterminate Form
When \( x \) approaches 0, the expression \( \frac{1}{x} \) approaches infinity, while \( \ln(1 + x) \) approaches \( \ln(1) = 0 \). Therefore, the limit is of the form \( \frac{0}{0} \), which is an indeterminate form.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule. To do this, we take the derivative of the numerator and the derivative of the denominator separately:

- **Numerator:** \( \ln(1 + x) \)
  - The derivative of \( \ln(1 + x) \) with respect to \( x \) is \( \frac{1}{1 + x} \).
  
- **Denominator:** \( x \)
  - The derivative of \( x \) with respect to \( x \) is 1.

Applying L'Hôpital's Rule gives:

\[
\lim_{x \to 0} \frac{\frac{1}{1 + x}}{1} = \lim_{x \to 0} \frac{1}{1 + x}
\]

### Step 3: Compute the Limit
Now, substitute \( x = 0 \) into the simplified expression:

\[
\frac{1}{1 + 0} = \frac{1}{1} = 1
\]

So, the limit is:

\[
\boxed{1}
\]

Would you like more details or have any other questions? Here are some related questions:

1. What are some other situations where L'Hôpital's rule is applicable?
2. Why is it important to check for indeterminate forms before applying L'Hôpital's rule?
3. How would this problem be solved without L'Hôpital's rule, using series expansion or another method?
4. Can L'Hôpital's rule be applied multiple times, and if so, in what situations?
5. What are the potential pitfalls when applying L'Hôpital's rule?

**Tip:** When dealing with logarithmic functions in limits, consider if using L'Hôpital's rule or expanding the logarithm can simplify the problem before diving into the calculation.

Learn how to solve the limit lim(x -> 0) (1/x * ln(1 + x)) using L'Hôpital's Rule. This problem involves logarithmic functions and demonstrates the application of L'Hôpital's Rule to resolve indeterminate forms. Suitable for college-level mathematics.

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