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How would you evaluate the following limit as n goes to infinity?

$$\lim \frac {1}{(1+\frac {1}{n})^n}$$

I would of thought that this would evaluated to be,

$$\lim \frac {1}{(1)^n} = 0 $$

However the correct answer is $$\frac{1}{e}$$

Arnaldo
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user123
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1 Answers1

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You can not answer like that because you are not allowed to seperate the limit in those two limits.

Hint: Use (the definition) $$\lim_{n\to \infty} \left( 1 + \frac{1}{n} \right)^n = \mathrm e.$$

  • so that definition should just be memorized and then it would be simply $\frac {1}{e}$ – user123 Feb 14 '17 at 20:58
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    Yes, if this is your definition of $\mathrm e$. –  Feb 14 '17 at 20:58
  • Well, it's one possible definition of $\mathrm e$. Another possible definition is $$\mathrm e=\sum_{n=0}^\infty \frac{1}{n!}.$$ When using this definition for $\mathrm e$, you certainly do have to prove that $\lim_{n\to\infty}(1+1/n)^n = \mathrm e$. – celtschk Feb 14 '17 at 21:05
  • @celtschk And yet another definition is the inverse function of $\int_1^x \frac1t ,dt$. But the OP has not provided any information to suggest a starting definition. – Mark Viola Feb 15 '17 at 04:11