Calculus Horizontal and Diagonal Asymptotes
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Graphing More Special Cases
Horizontal and Diagonal Asymptotes

 
 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) = 

3X5 + 2X2

5X5 - 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)

3X5 + 2X2 x 1

X5


5X5 - 6 1

X5
 Multiply through and simplify:

F(x)

3X5 + 2X2


X5 1 X5 3

5X5 - 6


X5 1 X5
 

F(x)

3 + 2

X3

5 - 6

X5
 
 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 X3
Xое

Xое

5 - 6

Lim X5
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) = 

4X5 + 3X2

2X4 + 2X3
 
 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:
 

F(X)

4X5

2X4
 
 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

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