I need to evaluate the limit of function y=(1-cos2x)/x^2, using trigonometric identities. x approaches to 0.

We'll use the half angle identity to
numerator:

1 - cos 2x = 2 (sin
x)^2

We'll substitute 1 - cos 2x by 2 (sin x)^2 and we'll
get the equivalent fraction:

(1-cos2x)/x^2 = 2 (sin
x)^2/x^2

We'll evaluate the limit of the function 2 (sin
x)^2/x^2, if x approaches to 0.

lim 2 (sin
x)^2/x^2

We'll create remarkable limit: lim (sin x)/x =
1

According to this, we'll
get:

lim 2 (sin x)^2/x^2 = 2lim (sin
x)^2/x^2

2lim (sin x)^2/x^2 = 2lim (sin x)/x*lim (sin
x)/x

2lim (sin x)^2/x^2 =2*1*1 =
2

The limit of the function y =
(1-cos2x)/x^2, if x approaches to 0, is: lim (1-cos2x)/x^2 =
2.