The greatest integer function is .

Graph :

Graph of the function is :

Observe the graph.

Left hand limit :

As  tends to  from left hand side,  approaches to .

.

Right hand limit :

As  tends to  from right hand side,  approaches to .

.

Left hand limit and right hand limit are equal, so limit exists.

.

From the graph .

So  exists but it is not equal to .

 exists but it is not equal to .