The greatest integer function is , .

(a)

Graph :

Graph the function  :

(b)

(i)

Find .

Observe the graph.

As  tends to  from left hand side,  approaches to .

.

As  tends to  from right hand side,  approaches to .

.

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

.

(ii)

Find .

Observe the graph.

As  tends to  from left hand side,  approaches to .

.

(iii)

Find .

Observe the graph.

As  tends to  from right hand side,  approaches to .

.

(iv)

Find .

.

.

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

 does not exist.

(c)

Find .

Observe the graph.

 exists in the interval  except .

(a) Graph the function  is

(b) (i) .

    (ii) .

    (iii) .

    (iv)  does not exist.

(c)  exists in the interval  except .