Expert: Paul Klarreich - 1/30/2008

Question
Prove that:
lim      (1-cos x)/(x) = 0
(x --> 0)

I know how to prove the first part of this theorem correctly but I don't know how to prove for this part. Help? thank you.

Answer
Questioner:   Al
Category:  Calculus
Private:  No
 
Subject:  Proof for a theorem
Question:  Prove that:
lim      (1-cos x)/(x) = 0
(x --> 0)

I know how to prove the first part of this theorem correctly but I don't know how to prove for this part. Help? thank you
...................................
Hi, Al,

Something bad happened.  I am sure that you must have sent me the first part of the theorem -- it makes no sense to talk about it without sending it -- but it must have gotten lost.  I will have to guess.

The first part said:
    sin x
lim  ------ = 1
x->0   x

Now we use it to compute:
    1 - cos x
lim  ---------
x->0    x
Rationalize:


    1 - cos x 1 + cos x
lim  --------- ----------
x->0    x      1 + cos x

Multiply:
     1 - cos^2(x)
lim  --------------
x->0  x(1 + cos x)

Use trig:

       sin^2(x)
lim  --------------
x->0  x(1 + cos x)

Rearrange:

     sin(x)   sin x
lim  -------  -----------
x->0    x     (1 + cos x)

Use that first part:
          sin x
lim  1  -----------
x->0    (1 + cos x)

Limitize:
    sin 0
  -----------
  (1 + cos 0)
Compute:

     0
  ------- = 0/2 = 0
  (1 + 1)