Question

Find limx→0 (1 − cos x)/(x)^2 given that limx→0 (sin x)/(x)= 1 ?

Find limx→0
(1 − cos x)/(x)^2
given that limx→0
(sin x)/(x)= 1.

I think you shoud use I`hôpital

Answer 1

Using the trigonometric identity:

`sin^2theta =(1-cos 2 theta)/2`

we have:

`(1-cosx)/x^2 = (2sin^2(x/2))/x^2 = 1/2 sin^2(x/2)/(x/2)^2`

so:

`lim_(x->0) (1-cosx)/x^2 = 1/2 lim_(x->0) (sin(x/2)/(x/2))^2 = 1/2`