What is the limit of (1 - 1/n)^n as n approaches infinity?

[TranscendArcu]

Homework Statement

The Attempt at a Solution

So I know that the limit as n → ∞ of (1 - \frac{1}{n})^n = \frac{1}{e}. Using this information, is it legitimate to observe:

The limit as n → ∞ of (1 - \frac{1}{n})^{n ln(2)} = the limit as n → ∞ of ((1 - \frac{1}{n})^n)^{ln(2)} = e^{-1 ln(2)} = e^{ln(\frac{1}{2})} = \frac{1}{2}

 

[Robert1986]

Looks good to me.

 

[Dick]

Homework Statement

The Attempt at a Solution

So I know that the limit as n → ∞ of (1 - \frac{1}{n})^n = \frac{1}{e}. Using this information, is it legitimate to observe:

The limit as n → ∞ of (1 - \frac{1}{n})^{n ln(2)} = the limit as n → ∞ of ((1 - \frac{1}{n})^n)^{ln(2)} = e^{-1 ln(2)} = e^{ln(\frac{1}{2})} = \frac{1}{2}

Looks ok to me.