In this question we will show that the derivative of g(x) =

cos(ax) is g 0

(x) = −a sin(ax), using the limit definition of the derivative, i.e., using g

0

(x) = lim h→0 g(x + h) − g(x)

h

.

a. First, consider the two functions G1(m) =

cos(m) − 1 m

and G2(m) =

sin(m)

m

.

Notice that G1(0) and G2(0) are undefined. Determine the values of the limits lim m→0 G1(m) and lim m→0 G1(m)

in two ways.

i. For each function, fill out the table below. What does this suggest about the value of the limit? Explain.

ii. For the second way, plot each function. What does this suggest about the value of the limit? Explain. What is misleading about these plots?

b. Now determine the values of the limits lim h→0 cos(ah) − 1 h and lim h→0 sin(ah)
h for some constant

a, by using the limits above and setting m = ah.

c. Using your answer to part

(b) and the definition of the derivative given above, show that the derivative of g(x) = cos(ax) is g 0 (x) = −a sin(ax).