Calculus Using Limits to Solve Indeterminate Trig Functions
Math-Prof HOME Calculus Table of Contents Ask A Question PREV NEXT

A Limit to Trig
Using Limits to Solve Indeterminate Trig Functions

 
  Trig functions can have limits applied to them just like any other function.
 
 To help get us started, we have 3 tricks that help simplify limits used on them.
 Officially, these puppies are called "Trig Limit Theorems"
 Math buzz words strike again!
  
 #1
 The first one is pretty obvious:
 In plain English it says:
 The limit of a sine or cosine function as the variable approaches some value
 is the trig function applied to that value.
 
  That is, if a stands for any number we might have. Then:
Lim

 Sin X = Sin a

and Lim

 Cos X = Cos a

X a X a
 
  This is maybe kind of a "duh!, " but to do it,
 we need an official rule that says we can.
 
 The next two are more interesting.
 Think of them as ways to cancel trig stuff out of your problem.
 The trick is to use algebra to arrange your problem to include one of them
 and ZAP! out goes the trig.
 
  #2
Lim Sin X = 1

X 0 X
 
 What this one says is that for really small angles 
 the sine is also a really small number
 and as they get smaller, they get closer to being the same value.
 Something over itself equals 1.
 
  #3
Lim 1 - Cos X = 0

X 0 X
 
  This one is kind of the same deal.
  But here, as the cosine gets close to 1 the angle gets close to zero
  but at a slower rate.
 
 So while both the numerator and denominator get close to zero, 
  the numerator gets there faster.
 
 That makes the limit of the fraction as x approaches zero equal to zero.
 
  Example:
Lim Sin X - Cos XSin X

X 0 X2 Cos X
  
  Factor the numerator:
Lim Sin X(1 - Cos X)

X 0 X2 Cos X
  
 Separating terms:
Lim ( Sin X x 1 - Cos X x 1 )



X 0 X X Cos X
  
  Distributing the "Lim"
 
Lim Sin X x Lim 1 - Cos X x Lim 1



X 0 X X 0 X X 0 Cos X
 
  Using #2 and #3:
 
1 x 0 x Lim 1

X 0 Cos X
 
  We COULD figure out the last part,
 but since we are multiplying three things together and one of them is a zero
 the rest usually won't matter .
 
  As long as the rest of the terms are defined, the answer will be zero.
 

   copyright 2005 Bruce Kirkpatrick

Math-Prof HOME Calculus Table of Contents Ask A Question PREV NEXT