To solve the limit $\lim_{x \to 0} (1 + x)^{\cot(x)}$, we'll proceed by first expressing the limit in a more convenient form.

Step 1: Rewrite the Expression

The given expression is $(1 + x)^{\cot(x)}$. We can rewrite this expression using the exponential function:

$(1 + x)^{\cot(x)} = e^{\cot(x) \cdot \ln(1 + x)}$

So, the limit becomes:

$\lim_{x \to 0} (1 + x)^{\cot(x)} = \lim_{x \to 0} e^{\cot(x) \cdot \ln(1 + x)}$

Step 2: Focus on the Exponent

Now, we need to evaluate the limit of the exponent $\cot(x) \cdot \ln(1 + x)$ as $x$ approaches 0.

$\lim_{x \to 0} \cot(x) \cdot \ln(1 + x)$

Recall that $\cot(x) = \frac{\cos(x)}{\sin(x)}$. Near $x = 0$, $\cot(x)$ tends to $\frac{1}{x}$ because $\sin(x) \approx x$ and $\cos(x) \approx 1$.

Thus:

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

Step 3: Apply L'Hôpital's Rule

The limit $\lim_{x \to 0} \frac{\ln(1 + x)}{x}$ is of the indeterminate form $\frac{0}{0}$, so we can apply 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} = 1 ]

Step 4: Final Calculation

So, the limit of the exponent is:

$\lim_{x \to 0} \cot(x) \cdot \ln(1 + x) = \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$

Thus:

$\lim_{x \to 0} (1 + x)^{\cot(x)} = e^1 = e$

Final Answer:

$\lim_{x \to 0} (1 + x)^{\cot(x)} = e$

Would you like more details on any part of this solution, or do you have any questions?

Here are some related questions to explore further:

1. What is the derivative of $\ln(1 + x)$?
2. How does the limit $\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$ relate to the concept of the derivative?
3. Can you apply L'Hôpital's Rule to other indeterminate forms?
4. How would you solve this limit without using L'Hôpital's Rule?
5. What happens if we change $\cot(x)$ to $\tan(x)$ in the original limit?
6. How does this limit relate to the number $e$?
7. What are some other common limits that result in $e$?
8. How does the behavior of $\cot(x)$ as $x$ approaches 0 affect this limit?

Tip: When evaluating limits, especially those involving exponentials or logarithms, consider rewriting the expression using known limit properties or applying L'Hôpital's Rule if you encounter an indeterminate form.