



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 








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 
X^{2
}Cos X 






Factor the
numerator: 


Lim 
Sin
X(1  Cos X) 

X
®
0 
X^{2
}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 
