lim⁡𝑥→0ln⁡(1+𝑥)𝑥lim*x*→0​*x*ln(1+*x*)


It looks like you're asking for the limit of the expression \(\lim_{x \to 0} \frac{\ln(1 + x)}{x}\). This limit is important in calculus and can be evaluated directly.

### Step 1: Consider the Limit
We are interested in evaluating the limit:

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

### Step 2: Recognize the Indeterminate Form
As \( x \) approaches 0, both the numerator \(\ln(1 + x)\) and the denominator \(x\) approach 0, which gives an indeterminate form of type \( \frac{0}{0} \). This suggests that we can apply L'Hôpital's Rule.

### Step 3: Apply L'Hôpital's Rule
L'Hôpital's Rule states that for a limit of the form \( \frac{0}{0} \), we can take the derivative of the numerator and the derivative of the denominator separately and then take the limit of the resulting expression.

1. The derivative of \( \ln(1 + x) \) with respect to \( x \) is \( \frac{1}{1 + x} \).
2. The derivative of \( x \) with respect to \( x \) is 1.

Applying L'Hôpital's Rule:

\[
\lim_{x \to 0} \frac{\ln(1 + x)}{x} = \lim_{x \to 0} \frac{\frac{d}{dx}[\ln(1 + x)]}{\frac{d}{dx}[x]} = \lim_{x \to 0} \frac{\frac{1}{1 + x}}{1} = \lim_{x \to 0} \frac{1}{1 + x}
\]

### Step 4: Evaluate the Limit
As \( x \) approaches 0, \( 1 + x \) approaches 1, so:

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

### Final Answer:
\[
\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1
\]

Would you like more information on this limit or have any additional questions?

Here are some related questions to consider:

1. Why does L'Hôpital's Rule apply to indeterminate forms like \( \frac{0}{0} \)?
2. Can this limit be solved using a Taylor series expansion?
3. How does this limit relate to the derivative of \( \ln(1 + x) \) at \( x = 0 \)?
4. What is the significance of this limit in defining the natural logarithm function?
5. How would this limit change if the expression inside the logarithm were \( (1 - x) \) instead of \( (1 + x) \)?
6. What are other common limits that are important in calculus?
7. Can this limit be derived using the definition of a derivative?
8. How does this limit help in understanding the behavior of logarithmic functions near zero?

**Tip:** When you encounter limits involving logarithms, consider using L'Hôpital's Rule or series expansions for a straightforward solution.

Learn how to evaluate the limit lim(x -> 0) ln(1 + x) / x using L'Hôpital's Rule and understand its significance in calculus. This limit explores the behavior of the natural logarithm function near zero.

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