a.

The difference quotient is .

Let .

.

difference quotient :  .

Apply sine sum identity : .

.

Thus, the difference quotient is .

b.

The function is .

Substitute in above function.

.

Draw a coordinate plane.

Graph the two function .

Graph :

The function is .

Substitute in above function.

Draw a coordinate plane.

Graph the two function .

Graph :

The function is .

Substitute in above function.

.

Draw a coordinate plane.

Graph the two function .

Graph :

The function is .

Substitute in above function.

.

Draw a coordinate plane.

Graph the two function .

Graph :

c.

Observe the graphs in part b.

As approaches zero, the graph look like the graph of the function .

a.

Difference quotient is .

b.

Graph of is :      

Graph of is :

Graph of :

Graph of is :

c. .