limit

First question:
With reference to the diagram below:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Mar08/Crazylim4.jpg

Second question: Again, it can be proved by the squeezing principle:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Mar08/Crazylim5.jpg

Third question: Existence of limit can be found by verifying the existence and consistency of left-hand and right-hand limits, for example:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Mar08/Crazylim6.jpg

So from the result, the limit of x - [x] does not exist for x tends to 2 since the left-hand and right hand limits are not equal.
In fact for x - [x], its limit does not exist for all x tends to any integer (either positive or negative).
Therefore, the limit of a function can be said to be not existing if:

Either left-hand or right-hand or both limits do not exist; OR
Both limits exist but they are NOT equal to each other