Question

lim x tends to 0 (1-cos^3x)/xsin2x

Answer 1

Lim( x→0) { 1- cos³x}/xsin2x

here ,
( 1 - cos³x) = (1 -cosx)(1 + cos²x + cosx)
= (2sin²x/2)( 1 + cos²x + cosx)
and use concept ,
Lim( x → 0) sinx/x = 1

now,

Lim ( x →0) (2sin²x/2/x²/4 × x²/4 )( 1 + cos²x + cosx)/x(sin2x/2x) ×2x

= ( 2× 1/4) ( 1+ 1 + 1)/( 2)

= 3/4 ( answer )