On-line Math 21

On-line Math 21

4.5  Curve Sketching

Asymptotes: An asymptote is a line, vertical, horizontal, or slanted, which the graph of the function approximates ``near infinity''. This was already discussed somewhat back in <a href="../Unit1/limits-infinite.html»Chapter 1</a>, when we were dealing with infinite limits.

More About Asymptotes

  1. A vertical asymptote is a vertical line x = a , where a is a point at which the function has an infinite limit. If

    lim
    x® a 
    		 f(x) = ¥,
    then x = a is a vertical asymptote for the graph. This holds for one-sided limits as well. Usually, but not always, vertical asymptotes are places where the denominator of the expression for f(x) has value 0 . f(x) = lnx also has a vertical asymptote at x = 0 .

  1. A horizontal asymptote is a horizontal line y = c , where

    lim
    x® ±¥ 
    		 f(x) = c.
    The limit could approach c at either +¥   -¥, or both.

  2. A slant asymptote is a situation where the graph y = f(x) approaches some other line as x® ¥ (or -¥). You can see when this occurs by finding when there is some line y = mx+b for which

    lim
    x® ¥ 
    		 ( f(x)-(mx+b)) = 0.

Roughly, you should look for situations where f(x) has an expression which is linear except for terms of lower order, like
f(x) = x3+2x+3
x2-x+1
 .
This function has a slant asymptote of y = x+1 , since

lim
x® ¥ 
 ( f(x)-(x+1))
=

lim
x® ¥ 
 x3+2x+3-(x+1)( x2-x+1)
x2-x+1
=

lim
x® ¥ 
 x3+2x+3-( x3+x2-x2-x+x+1)
x2-x+1
=

lim
x® ¥ 
 2x+2
x2-x+1
=
0.

Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 21 Dec 2000, 00:39.