Find the limit?

Question

lim .. cos x - 1
x->0 ------------
.......... sin x



lim .. sin(cosx)
x->0 ------------
.......... sec x

Answer 1

The first limit answer is 0 and the second one is sin(1)
but how:
if you have cosx when x-->0 you can replace it to 1-x^2/2 and if you have sinx when x-->0 you can replace it to x
as you can see if you do so it happens that the first limit turn to 0
there is another way too about limit with 0/0 or inf/inf situation
you should differentiate cosx-1 and sinx seperatively and answer will be -sinx and cosx
now try this limit(-sinx/cosx) x-->0
and about the second one
cos(0)=1
sec(0)=1
sin(1)=sin(1)
so it's easy.
Sorry for my English.

Answer 2

1. Multiply top and bottom by sin x. The bottom becomes sin^2(x) Convert to 1-cos^2(x) Factor then cancel the cosx-1 Answer = 0


2. Sin(1) is the answer.