I got stuck on this one, once i saw the sin, i just didn't know what to do. All this is one problem. The Answer according to the book is Cos(x)
f(x) = sin(x)
f(x) = lim sin(x+h) - sin (x) over h
........h->0
Interesting, let me give this a bash:
$\displaystyle \frac{f(x+h)-f(x)}{h}=\frac{sin(x+h)-sin x}{h}$
$\displaystyle \frac{sin x cos h+sin h cos x-sinx}{h}$
as h---->0:
$\displaystyle sin h\rightarrow h \, and \, cos h\rightarrow 1$
$\displaystyle \frac{sin x+hcos x-sinx}{h}$
$\displaystyle \frac{hcos x}{h}$
$\displaystyle =cosx$