Proof of: lim_q→0 (cosq -1)/q = 0

We will begin by doing the following,

lim_q→0 (cosq -1)/q = lim_q→0((cosq - 1)(cosq + 1))/(q (cosq + 1))

= lim_q→0(cos²q - 1)/ (q (cosq + 1))                                           (7)

Here, let's recall that,

cos²q + sin²q = 1      =>   cos²q - 1 = -sin²q

By using this in (7) provides us,

= lim_q→0(sin²q)/ (q (cosq + 1))

= lim_q→0 (sinq/q)(-sinq)/(cosq + 1)

= lim_q→0 (sinq/q)  lim_q→0 (-sinq)/(cosq + 1)

Here, as we just proved the first limit and the second can be got directly we are pretty much done.  All we require to do is get the limits.

lim_q→0 cosq - 1

= lim_q→0 (sinq/q)  lim_q→0 (-sinq)/(cosq + 1)

= (1) (0)

 = 0