Expert: Paul Klarreich - 4/30/2009

Question
Hi, So I have a worksheet that is just to help us get started before we start our next lesson and I was just really confused on it because I can't get one part of the sheet without getting something else. So anything helps!

1. Find lim   cos(theta)-1
  (theta)->0 ------------
               (theta)
using the following outline:

a) multiply by cos(theta)+1
              ------------
              cos(theta)+1


b) Use the Pythagorean identity to simplify the numerator


c) rewrite the fraction as a product of two fractions, where one of the fractions is sin(theta)
                             ----------
                              (theta)


d) use limits laws to find  lim      cos(theta)-1
                         (theta)->0 ------------
                                       (theta)  

Answer
Questioner:   ray
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  just some randoms
Question:  Hi, So I have a worksheet that is just to help us get started before we start our next lesson and I was just really confused on it because I can't get one part of the sheet without getting something else. So anything helps!

1. Find lim   cos(theta)-1
 (theta)->0 ------------
              (theta)
using the following outline:

a) multiply by cos(theta)+1
             ------------
             cos(theta)+1


b) Use the Pythagorean identity to simplify the numerator


c) rewrite the fraction as a product of two fractions, where one of the fractions is sin(theta)
                            ----------
                             (theta)


d) use limits laws to find  lim      cos(theta)-1
                        (theta)->0 ------------
                                      (theta)
..........................................
Hi, Ray,

This is a standard derivation found in most calc texts.  You are going to use this limit fact:
       sin t
lim   -------- = 0
t->0      t

{BTW, I will use  t for theta to save typing.]


Find
lim   cos(t)-1
t->0  ------------
        t
using the following outline:

a) multiply by cos(t)+1
             ------------
              cos(t)+1
.........................

lim   cos(t)-1  cos(t)+1
t->0  --------- ---------
        t      cos(t)+1


lim     cos^2(t)-1
t->0  --------------
      t (cos(t)+1)


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

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

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

..........................
lim    -  sin(t)  lim       sin(t)
t->0  ----------  t->0   -----------
          t              cos(t) + 1
<<< first part >> << second part >>

OK, the first part --> -1 because of that earlier 'fact'.

The second part --> 0/1 = 0

So the product is zero.