2. Theorem method:

(1) monotone bounded sequence must have a limit;

(2) Pinch criteria;

(3) Mathematical induction (possibly combining (1) and (2))

3. Function method: the general term formula of the series is formed into a function, and the limit of the series is determined by finding the limit of the function, which should be used together with pinching criterion or conceptual method.

1, prove the existence of the limit of the sequence {xn=(n- 1)/(n+ 1)} and find its limit.

Prove:

∫ 1- 1/( 1+ 1/n)= 1-n/(n+ 1)& lt； 1-2/(n+ 1)= xn & lt； (n- 1)/n = 1- 1/n

Namely:1-1(1+1/n) < xn < (n- 1)/n = 1- 1/n

It is known that lim 1/n =0 when n is infinite.

∴lim[ 1- 1/( 1+ 1/n)]= 1

lim[ 1- 1/n]= 1

According to the clamping center: xn limit exists, limxn= 1.