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I tried to use the squeezing principle, but couldn't find a proper expression for the left inequality. Maybe there's another way, it seems simple though I couldn't figure it out.

$$? \le \mathop {\lim }\limits_{n \to \infty } {{n - 1} \over {n + 1}} \le \mathop {\lim }\limits_{n \to \infty } {{n + 1} \over {n + 1}} = 1$$

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$$\frac{n-1}{n+1}=1-\frac2{n+1}$$ –  lab bhattacharjee Dec 1 '13 at 11:09
    
The tag (limit-theorems) is intended for questions about limit theorems in probability theory and not for questions about determining limits of sequences or functions, see the tag-wiki and the tag-excerpt. (The tag-excerpt is also shown when you are adding a tag to a question.) –  Martin Sleziak Dec 1 '13 at 14:29

3 Answers 3

up vote 5 down vote accepted

Why not to show it directly: $$\mathop {\lim }\limits_{n \to \infty } {{n - 1} \over {n + 1}}=\mathop {\lim }\limits_{n\to \infty } {{1 - \frac1n} \over {1 + \frac1n}}=1$$

This inferred from theorems for limit of quotient and limit of difference/sum.

EDIT:You can use the limit definition:

You need to show that $$(\forall\epsilon>0)(\exists n_0>0)(\forall n>n_0):\left|{{n - 1} \over {n + 1}}-1 \right|<\epsilon$$ $$(\forall\epsilon>0)(\exists n_0>0)(\forall n>n_0):\left|{{n - 1 -(n+1)} \over {n + 1}} \right|<\epsilon$$ $$(\forall\epsilon>0)(\exists n_0>0)(\forall n>n_0):{|{-2}| \over |{n + 1}|} <\epsilon$$ $$(\forall\epsilon>0)(\exists n_0>0)(\forall n>n_0):{2 \over {n + 1}} <\epsilon$$ $$(\forall\epsilon>0)(\exists n_0>0)(\forall n>n_0):{2 \over \epsilon} <{n+1}$$ Take $n_0=\lceil 2/e \rceil$ and the ineqality holds.

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Oops. I got to wake up before doing math –  Daniel Gagnon Dec 1 '13 at 11:12
    
By the way, what happens for the limit $\mathop {\lim (}\limits_{n \to \infty } {{n - 1} \over {n + 1}}{)^{{n^2}}}$ –  Daniel Gagnon Dec 1 '13 at 11:15
    
@DanielGagnon Intuitively the larger power in either the numerator or the denominator dominates the other. Your last ratio goes to infinity. –  JohnK Dec 1 '13 at 11:17
    
But how can you show it formaly? –  Daniel Gagnon Dec 1 '13 at 11:28
    
I use theorems of kind $\lim \frac ab=\frac{\lim a}{\lim b}$ and $\lim a+b=\lim a+\lim b$ –  LeeNeverGup Dec 1 '13 at 11:43

$$\lim_{n\to\infty}\frac{n-1}{n+1}=\lim_{n\to\infty}\frac{n\left(1-\frac{1}{n}\right)}{n\left(1+\frac{1}{n}\right)}=\lim_{n\to\infty}\frac{1-\frac{1}{n}}{1+\frac{1}{n}}=\frac{1-\lim_{n\to\infty}\frac{1}{n}}{1+\lim_{n\to\infty}\frac{1}{n}}=\frac{1-0}{1+0}=1$$

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How about L'Hopital's rule? After all when $n$ goes to infinity you have an indefinite ratio $\frac{\infty}{\infty}$ since the constants are negligent.

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