Can I Improve my answer?

Question

lim x -> inf [ (1-cosxcos2xcos3x) / (1-cosx) ]

(x approaches to +infinity)

My answer: = (1 - cos(pi).cos(2pi).cos(3pi)) / (1 - cos(pi)) = (1 - [(-1).1.(-1)] ) / (1 - (-1)) = (1 - 1) / (2) = 0/2 = 0

Hello, I have just put "pi" where I have seen X's and, the end result was 0/2 = 0, is this a correct way to solve this limit? Are there any other methods?

Answer 1

This limit DNE.

I think that you have lim x→0 then you have to apply L'Opital rule because you have 0/0 indeterminate form.

lim x→0 of (1 - cosxcos2xcos3x)/(1- cosx) = lim x→0 of +(1 - cosxcos2xcos3x)'/(1- cosx)' = (sinxcos2xcos3x + 2cosxsin2xcos3x + 3cosxcos2xsin3x)/sinx = lim x → 0 of [cos2xcos3x + 4cos²xcos3x + 3cosx(3-4sin²x)cos2x] = 1 + 4 + 9 = 14.

I use formulas:

sin2x = 2sinxcosx;

sin3x = 3sinx - 4sin³x