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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 $$\frac{1}{e}$$

1 Answers 1

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

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    so that definition should just be memorized and then it would be simply $\frac {1}{e}$ – 2017-02-14
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    Yes, if this is your definition of $\mathrm e$. – 2017-02-14
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    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 $\lim_{n\to\infty}(1+1/n)^n = \mathrm e$. – 2017-02-14
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    @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. – 2017-02-15