1. ## continuity with limits!

to prove that sine is continuous we need to show that limx--> sin(x)=sin(a) for every real number a. if we let h=x-a, then x=a+h and x-->a <--> h-->0. so an equivalent statement is that

limh-->0sin(a+h) = sin(a)

use the theorem:

limx-->0cos(x) = 1 and limx-->0sin(x)=0

to show that this is true.