(Pure Mathematics)How to prove a limit does not exist?

根據定義﹐若果一個函數f(x)沒有在某一點上沒有limit﹐則只有2個可能性
1 函數在該點c上的limit是無限大。
如lim(x→0) 1/x
2 可以找到兩個不同的序列﹐該2個序列皆趨向c﹐但f(x)趨向不同的值。
3 還有一個方法﹐就是找到一個序列﹐使該序列趨向c(| x -c| < δ)﹐但|f(x)-L|< ε 不成立。
但2同3只是一件事的兩種不同說法﹐所以以下只舉個用2的例子。3的方法請看參考資料
以下這條是經典例子
Prove that lim[x -> 0] sin(1 / x) does not exist.
Suppose the limit did exist, then there would be an L such that given an ε＞0,then |x| ＜ σ such that |sin(1/x)-L| ＜ε

let x1=1/(Nπ) such that |x1| ＜ σ

|sin(1/x1)-L|=|L|＜ε
now let x2=1/[(2n+1/2)π] such that |x2| ＜ σ

|sin(1/x2)-L|=|1-L|＜ε

but if take ε＜1/2, L will be close to both 0 and 1,a contradiction.

lim(x→0)(sin1/x) does not exist
http://www.math.washington.edu/~conroy/general/sin1overx/
這個網址繪出了sin1/x的圖﹐從圖看到sin1/x在x=0附近會在-1與1之間上下波動﹐因而亦可看出沒有limit
以下是另一條類似題﹐可以作為練習
Prove that the limit of f(x)={x, when x is rational; 0, when x is irrational} as x approaches c for ALL (c not equal to zero, c is real number) does not exist?