https://file.aiquickdraw.com/mathbot/file/17300342878983mr2jvrs.jpg

The problem given in the image can be analyzed in two parts:

1. **Interpreting the function**: We have a function \( \alpha^{f(z)} \), which is represented as \( e^{\alpha \ln(f(z))} \). This indicates that the power function \( \alpha^{f(z)} \) is expressed in terms of the exponential and logarithmic forms. This setup often prepares us for limits involving exponential growth or logarithmic properties.

2. **Evaluating the limit**: The limit provided is:

   \[
   \lim_{n \to \infty} \left( 1 + \frac{i}{n} \right)^n
   \]

   This type of expression resembles the well-known limit definition of the exponential function. Specifically, if we have \( \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = e^x \), then applying this property with \( x = i \) would give:

   \[
   \lim_{n \to \infty} \left( 1 + \frac{i}{n} \right)^n = e^i
   \]

   Using Euler's formula, \( e^{i} = \cos(1) + i \sin(1) \), the answer to the limit is:

   \[
   e^{i} = \cos(1) + i \sin(1)
   \]

Thus, the final answer is:

\[
\cos(1) + i \sin(1)
\]

Would you like a detailed explanation of each step?

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Here are some related questions to expand your understanding:

1. What is Euler's formula, and how does it apply to complex exponentials?
2. How does the limit \( \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = e^x \) work, and why?
3. What is the significance of expressing powers as exponential and logarithmic forms?
4. How do complex numbers and the exponential function relate in trigonometric terms?
5. Can this technique apply to other complex limits, and if so, how?

**Tip:** Remember, limits of the form \( \left( 1 + \frac{x}{n} \right)^n \) often lead to exponential expressions when \( n \to \infty \).

设幂函数 α^(f(z)) 取 e^(α ln(f(z))) 的分支，则极限 lim(n→∞)(1 + i/n)^n = (  ).

Explore the solution to the complex limit problem: lim(n→∞)(1 + i/n)^n. This problem uses Euler's formula to express complex exponentials and evaluates the limit as n approaches infinity. Suitable for college-level students.

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