Roman C. answered • 01/30/17

Masters of Education Graduate with Mathematics Expertise

You have the correct first step. Here is the full derivation.

We know that lim

_{x→0}(sin x)/x = 1. This limit is a classic exercise in using the Squeeze Theorem.Also,

lim

_{x→0}(1 - cos x)/x= lim

_{x→0}(1 - cos^{2}x)/[x(1 + cos x)]= lim

_{x→0}(sin^{2}x)/[x(1 + cos x)]= [lim

_{x→0}(sin x)/x][lim_{x→0}(sin x)/(1 + cos x)]= 1·0

= 0

Now use your reduction.

lim

_{x→0}(cos(x + π/3) - 1/2)/x= lim

_{x→0}(cos x cos π/3 - sin x sin π/3 - 1/2)/x= lim

_{x→0}((1/2)cos x - (√3/2)sin x - 1/2)/x= (-√3/2)lim

_{x→0}(sin x)/x - (1/2)lim_{x→0}(1 - cos x)/x= -√3/2