(a)

The symbol  indicates the greatest integer function.

The greatest integer function is also called as floor function.

Graph :

Graph of the function :

(i)

Find .

Observe the graph.

As  tends to  from right hand side, the value of  approaches to .

.

(ii)

Find .

Observe the graph.

As  tends to  from right hand side, the value of  approaches to .

.

As  tends to  from left hand side, the value of  approaches to .

.

Left hand limit and right hand limit are not equal, so limit does not exist.

 does not exist.

(iii)

Find .

Observe the graph.

As  tends to  from right hand side, the value of  approaches to .

.

As  tends to  from left hand side, the value of  approaches to .

.

Left hand limit and right hand limit are not equal, so limit exist.

.

(b)

(i)

Find .

Observe the graph.

If  tends to number from left hand side, then  approaches to next lower number.

.

(ii)

Find .

Observe the graph.

If  tends to number from right hand side, then  approaches to same number.

.

(c)

Find .

Observe the graph.

 exists when  is not an integer.

(a) (i) .

    (ii)  does not exist.

    (ii) .

(b) (i) .

     (ii) .

(c)  exists when  is not an integer.