means that the graph of f(x) has a horizontal asymptote on the right at y = L. 2.4 Limits at Infinity
means that the graph of f(x) has a horizontal asymptote on the right at y = L.
means that the graph of f(x) has a horizontal asymptote on the left at y = L.
A graph has a horizontal asymptote at L if, for any tiny vertical band about y=L, eventually the
graph is within that band and stays within the band.
Theorem: For r>0, c real,
and
.
For rational functions, there are 3 cases. Consider
, where p(x) and q(x) are
polynomials.
1. If deg p < deg q, the limit is 0.
2. If deg p = deg q, the limit is the ratio of the leading coefficients.
3. If deg p > deg q, the limit is
or
, depending on the sign of the expression.
Note that for rational functions, if there is an asymptote on one side (right or left), then there is
the same asymptote on the other side. This is not true for many other functions, e.g. arctan x.
For rational functions, the limit can be evaluated by ignoring all terms except the dominant term
on top and bottom. The dominant term is the term of highest degree.
Examples:
1. 
=
2. 
=
3. 
=
