



If the
right side of your equation is a fraction



where the
biggest exponent on top of the fraction is the same power 


as the
biggest exponent on the bottom of the fraction, 


the
equation has a horizontal (side to side) asymptote. 





But this
one doesn't happen at Y = 0 





To find
out where it happens, you throw out everything in the problem 


except
for the numbers next to the largest power of X. 


That is,
the coefficients. 





If you
don't see a number there (like in, say X
^{4}) 


the
number is 1 





The
fraction that these two coefficients make 


is the
place where the horizontal asymptote lives. 





Example: 








To find
the asymptote, throw away everything 


except
the numbers next to the biggest power of X. 











There is
one other way that we can have a horizontal asymptote. 





This
happens when the biggest exponent on the bottom of the fraction 


is bigger
than the biggest exponent on the top of the fraction. 


Like
this: 








When this
happens, the horizontal asymptote 


is at Y =
0. 





Let's
graph this one ... 


X 
Y 
10 
.005 
5 
.016 
3 
.034 
1 
0.25 
0 
3 
1 
0 
3 
.051 
5 
.020 
10 
.005 















Another
interesting thing happens when the biggest exponent on top 


is one
more than the biggest exponent on the bottom. 


Like this
... 








What we
do here, is to get rid of all of the smaller exponent terms. 


Just
leave the term with the biggest exponent in the numerator 


and also
leave the term with the biggest exponent in the denominator, 


like this
... 








Now just
simplify ... 








So the
asymptote happens at the line Y = 2X 











Let's
graph the equation. 


Start
with a T table ... 


X 
Y 
10 
19.8 
5 
9.61 
3 
3.17 
1 
1.25 
0 
0.333 
1 
1.75 
3 
5.75 
5 
9.68 
10 
19.8 















In that
last one, the biggest exponent on top 


was one
bigger than the biggest exponent on the bottom. 





What do
you think would happen if the biggest exponent on top 


was 2
bigger than the biggest exponent on the bottom?? 


Say,
something like this ... 








If we use
the same trick we've been using, 


we would
get rid of everything but the biggest exponent term 


in the
numerator and the biggest exponent term in the denominator. 


Like this
... 





and then
we simplify ... 





and graph
... 





Is this a
real asymptote? 





It
depends on who you ask. 





Some
people say that asymptotes have to be straight lines. 


But if
you graph this complex equation, 


the graph
line keeps getting closer to the line Y = 3X
^{2} 


So it
works just like any other asymptote. 





I say: 


If it
walks like an asymptote, 


and
quacks like an asymptote, 


then ... 


It's an
AFLAC commercial. 





copyright 2005 Bruce Kirkpatrick 
