# Two tricky limits - which theorems should I use?

I have to find a limit as $n\rightarrow\infty$ of 2 sequences:

$\lim\space (0,9999+\frac{1}{n})^n$

$\lim\space (1,00001-\frac{1}{n})^n$

Intutition tells me that as n goes to infinity $\frac{1}{n}$ becomes so small we can throw it out of the equation and it all comes down to finding limits of $0,9999^n$ and $1,00001^n$ which is trivial. But $\lim\space (1+\frac{1}{n})^n=e$ shows that this intuition may be wrong. So what should I do about these limits in order to prove them formally? Which theorems could be useful?

• For a intuitive notion, please see this – cjferes Oct 30 '14 at 12:25

## 2 Answers

Your intuition is right. To make it formal, find a lower and upper bounds: $$\left(0.9999+\frac 1n\right)^n\le0.99995^n$$ for sufficiently high $n$ and similarly for the other limit.

• Why is that an upper bound? – qiubit Oct 30 '14 at 12:50
• @user36346 When $n$ is sufficiently large, we have $\frac1n\le0.00005$ and you only need the inequality to hold for sufficiently large $n$ as the limit doesn't depend on finitely many members. – user2345215 Oct 30 '14 at 12:54

For $\;M\in\Bbb N\;$ such that $\;\frac1n=r<1-0.9999=0.0001\;\;\forall\, n>M$ , you get

$$0\le\left(0.9999+\frac1n\right)^n\le r^n\xrightarrow[n\to\infty]{}0$$

Try something similar with the second one.