Calculus Limits help pleez?

Question

Lim 1- Cos x

x->0 ------------- = 1/2

.......... x^2

(Ignore the dots)

if you could explain the specific rule behind it that would be great, i have the answer i just don't know how to get it in the future thanks my lecture being cut short due to a fire alarm but it's still due tomorrow awesome

Answer 1

Let s = lim (x -> 0) { [ 1 - cos(x) ] / x^2 }

Multiplying top and bottom by 1 + cos(x):

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

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

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

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

= { lim (x - > 0) [ sin(x) / x ] }^2 / 2

= 1^2 / 2

= 1 / 2.