lim x menuju alfa sinx-sin alfa/x-alfa

Lim (sin x - sin a)/(x - a)
= Lim (2 cos 1/2 (x + a) sin 1/2 (x - a)) / (x - a)
= Lim 2 cos 1/2 (x + a) . (sin 1/2 (x - a))/(x - a)
= 2 cos 1/2 (a + a) . 1/2
= 2 . cos a . 1/2
= cos a

sin x  - sin α = 2 cos ¹/₂ (x+α) sin ¹/₂(x-α)

limit (x →α)  (sin x - sin α)/(x- α)
= lim (x→α)  { 2cos ¹/₂ (x+α) sin ¹/₂ (x - α)}/{ x-α)
= lim(x→α)  2 cos ¹/₂(x+α) . lim(x→a) sin ¹/₂(x-α)/(x-α)
=  (2 cos α )(1/2)
= cos α