Expert: Paul Klarreich - 10/7/2005

Question
           cos(Pi/2 +h)- cos(Pi/2)
lim       __________________________
h->0                  h  

Answer
Hi, Stephen,
(Oops, you are Mr. Bubbles.  Anyway, this the answer I sent him.)

One of the difficulties I always face around this time of the year (early October) is that I know you are taking Calc I, but I don't know which week it is, and it depends on context, so I have to make a guess as to what your instructor wants. (who, I suppose, is a disembodied spirit at the other end of an Internet connection. By the way, your link didn't work -- you probably have a password or something. No problem, though -- I understand the question.)

[VIEW THIS IN A FIXED FONT]

Context I - You never heard of a derivative.

You want to find:
   cos(pi/2 + h) - cos(pi/2)
lim -------------------------
h->0 h

You will use these facts:
      sin x
1) lim ----- = 1, already proved, I hope.
  x->0   x

2) cos(A + B) = cos A cos B - sin A sin B

3) cos(pi/2) = 0
sin(pi/2) = 1

Using (2) on the top, you write:

    cos(pi/2) cos h - sin(pi/2) sin h - cos(pi/2)
lim ----------------------------------------------
h->0                        h

Now use (3)

    (0) cos h - (1) sin h - (0)
lim ----------------------------
h->0            h

   - sin h
lim ------- = -1
h->0    h

Context II - You have learned of derivatives.

You recognize the expression as the derivative of cos x at x=pi/2, so the answer should be - sin(pi/2) = - 1

[Yes, things get easier when you know more stuff.]