vertical asymptotes are associated with limits which become infinite , whereas horizontal asymptotes are associated with limits as a function's argument approaches infinity .

Definition: lim _{x-> infinity} f(x) = L
means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large

This is often said as "f(x) asymptotically approaches L as x approaches infinity."

Definition: L is a horizontal asymptote of f(x)
if

lim _{x-> - infinity} f(x) = L or lim _{x-> + infinity} f(x) = L

Negative powers of x asymptotically approach 0, as x-> infinity

lim _{x-> infinity} x ^-r = 0

Note: a function can have different horizontal asymptotes, as well as more than one vertical asymptote.

Knowledge of a functions asymptotes (both vertical and horizontal) is one of several key pieces of information, needed to graph the function accurately.

Definition: lim _{x -> infinity} f(x) = + infinity
if for some positive number M, there is a number N>0 such that f(x)>M if x>N.

1.

with(plots):
p1 := plot( (x^2-1)/(x^2+1), x=-20..20):
p2 := plot( 1, x=-20..20):
display([p1,p2]);

plot( (3*x^2-x-2)/(5*x^2+4*x+1), x=-20..20 );

plot( sqrt(2*x^2+1)/(3*x-5), x=-20..20 );

Last modified Wed Sep 18 20:59:23 CDT 1996