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}(cos^{2}q - 1)/ (q (cosq + 1)) (7)
Here, let's recall that,
cos^{2}q + sin^{2}q = 1 => cos^{2}q - 1 = -sin^{2}q
By using this in (7) provides us,
= lim_{q}_{→0}(sin^{2}q)/ (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
= (1) (0)
= 0