Verified answer

L = lim( x -> 0) [ 1 - cos^2 ( 2x) ] / x^2

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

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

= 2 [ lim ( sin 2x / 2x ) ]. 2 [ lim ( sin 2x / 2x ) ]

= 2 [ 1 ]. 2 [ 1 ]......lim sin( theta)/ ( theta) = 1

= 4. ...................................................................Ans.

lim {x->0} ( 1-cos²(2x))/x²

= lim {x->0} ( sin²(2x))/x²

= lim {x->0} (4 sin²(2x))/(2x)²

= 4