Can anyone help with a limit question containing cosine?

Question

I need help finding the limit of x as it approaches 0 for (cos(x+(pi/3))-(1/2))/x. 

I feel that the first step is reducing the equation to: ((cosx*cos(pi/3) - sinx*sin(pi/3))/x) -1/2x

Answer 1

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 - cos² x)/[x(1 + cos x)]

= lim_x→0 (sin² 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