



OK, 


When we
have a zero point in the denominator that can't be factored out 


we
have a vertical asymptote. 





There can
be horizontal asymptotes too. 


(Yeah I
know, big surprise given the name of this page) 





Horizontal
asymptotes happen 


when the highest power of the variable in the
numerator 


is the
same as the highest power of the variable in the denominator. 


Something
like: 


F(x)
= 
3X^{5}
+ 2X^{2} 

5X^{5}
 6 






When the
highest powers in the numerator and denominator are the same, 


there will be a
horizontal asymptote at the fraction 


made by the coefficients of the
highest power terms. 


In this
case, that will be at ^{3}/5. 





This is
because as X gets very big, 


the terms
with lower powers of X, and constants, get less and less important. 





This
probably makes sense just as it is, 


but math
types LOVE proofs. 


So here's
a proof of that 


"as X gets bigger and bigger" horizontal
asymptote thing. 


We call
it a limit as X approaches infinity. 





As usual
with these proofs, you can blow this one off if you want. 


Just
scroll down till I tell you it's over. 


But
really, this one's not that bad ... 





Here
goes. 


Start
with our original equation and multiply by a messy name for 1. 





F(x)
= 
3X^{5}
+ 2X^{2} 
x 
1 

X^{5}



5X^{5}
 6 
1 

X^{5}




Multiply
through and simplify: 


F(x)
= 
3X^{5}

+ 
2X^{2} 






X^{5}
^{1} 
X^{5
}^{3}






5X^{5}

 
6 






X^{5}
^{ 1} 
X^{5}









F(x)
= 
3 
+ 
2 





X^{3}






5 
 
6 





X^{5}









Now take
the limit of this as X goes to infinity. 


Remember,
the limit only cares about terms with X in them... 





Lim 
F(x)
= 
3 
+ 
2 





Lim 
X^{3}



Xое 



Xое 
5 
 
6 





Lim 
X^{5}



Xое 






Lim 
F(x)
= 
3
+ 0 
= 
3 


Xое 
5
 0 
5 






OK! 
Proof's
Over! 
Come
back now! 






The thing
that makes horizontal asymptotes different from vertical asymptotes 


is that
the graph line can cross a horizontal asymptote. 


It can't
cross a vertical asymptote. 








Diagonal
Asymptotes happen when the biggest exponent in the numerator 


is 1
larger than the biggest exponent in the denominator. 


For
example: 





F(X)
= 
4X^{5}
+ 3X^{2} 

2X^{4}
+ 2X^{3} 






When this
happens, 


the asymptote will be a diagonal line through the origin
((0,0) point) 


with a
slope equal to the coefficients of the largest X powers 


in the
numerator and the denominator. 





So in the
example above, the slope will be 2 (that is ^{4}/2). 





This
works about the same way as the horizontal asymptote. 


As the
value of X gets bigger, the terms with smaller powers get less
important. 


If we
thought of the equation as just the highest power term in the
numerator 


and the
highest power term in the denominator, it would be: 











And this
puppy simplifies to: 


F(X) = 2X






And this
is actually known as slope intercept form (F(X) = mX + b) 


Where m
is the slope and b is the intercept. 





OK, so m
(the slope) is 2, where's b? 





It's
zero. 


Remember,
we said that the diagonal asymptote when through the origin (0,0). 


That's
the intercept! 





copyright 2005 Bruce Kirkpatrick 
