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Chapter 7: Problem 6

Evaluate \(\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n}\)

Short Answer

Expert verified
The short version of the answer is: \[ \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n} = e^{-1} \text{ or } \frac{1}{e}. \]

Step by step solution

01

Identify the given expression and the limit definition

The given expression is: \[ \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n} \] The limit definition of the exponential function states that for any real number x, \[ \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = e^x \]
02

Compare the given expression with the limit definition

We can observe that if we replace x with -1 in the limit definition, we get our given expression in the same form. Therefore, we can write the given expression as: \[ \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n} = \lim _{n \rightarrow \infty}\left(1+\frac{-1}{n}\right)^{n} \]
03

Apply the limit definition of exponential function

According to the limit definition of the exponential function, we know that \[ \lim_{n \to \infty} \left( 1 + \frac{-1}{n} \right)^n = e^{-1} \]
04

Simplify the result and state the final answer

We have found the desired limit based on the limit definition of the exponential function: \[ \lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{n} = e^{-1} \] Thus, the final answer is \( e^{-1} \) or \(\frac{1}{e} \).

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