Evaluate `lim_(x to 0) (1 -cos x)/(x^(2))`

Explanation

We have, `underset(x to 0)(to) (1 - cos x)/(x) = underset(x to 0)(lim) (2 sin^(2) x//2)/(x^(2)) = 2 underset(x to 0)(lim) (sin x//2 .sin x//2)/(x)`
`= 2 underset(x to 0)(lim) (sin (x)/(2))/(2.(x)/(2)) underset(x to 0)(lim) (sin (x)/2)/(2.(x)/(2))`
`= (2)/(4) underset(x to 0)(lim) (sin(x)/(2))/((x)/(2)) underset(x to 0)(lim) (sin(X)/(2))/((x)/(2)) = (1)/(2) 1.1 = (1)/(2)`