Let’s consider several classes of functions here and look at the different types of end behaviors for these functions.
End behavior for polynomial functions
Consider the power function
where
is a positive integer. From
[link] and
[link] , we see that
and
For power functions with an even power of
For power functions with an odd power of
and
Using these facts, it is not difficult to evaluate
and
where
is any constant and
is a positive integer. If
the graph of
is a vertical stretch or compression of
and therefore
If
the graph of
is a vertical stretch or compression combined with a reflection about the
-axis, and therefore
If
in which case
Limits at infinity for power functions
For each function
evaluate
and
Since the coefficient of
is
the graph of
involves a vertical stretch and reflection of the graph of
about the
-axis. Therefore,
and
Since the coefficient of
is
the graph of
is a vertical stretch of the graph of
Therefore,
and
We now look at how the limits at infinity for power functions can be used to determine
for any polynomial function
Consider a polynomial function
of degree
so that
Factoring, we see that
As
all the terms inside the parentheses approach zero except the first term. We conclude that
For example, the function
behaves like
as
as shown in
[link] and
[link] .
The end behavior of a polynomial is determined by the behavior of the term with the largest exponent.
A polynomial’s end behavior is determined by the term with the largest exponent.
End behavior for algebraic functions
The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In
[link] , we show that the limits at infinity of a rational function
depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of
appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of
Determining end behavior for rational functions
For each of the following functions, determine the limits as
and
Then, use this information to describe the end behavior of the function.
(Note: The degree of the numerator and the denominator are the same.)
(Note: The degree of numerator is less than the degree of the denominator.)
(Note: The degree of numerator is greater than the degree of the denominator.)
The highest power of
in the denominator is
Therefore, dividing the numerator and denominator by
and applying the algebraic limit laws, we see that
Since
we know that
is a horizontal asymptote for this function as shown in the following graph.
The graph of this rational function approaches a horizontal asymptote as
Since the largest power of
appearing in the denominator is
divide the numerator and denominator by
After doing so and applying algebraic limit laws, we obtain
Therefore
has a horizontal asymptote of
as shown in the following graph.
The graph of this rational function approaches the horizontal asymptote
as
Dividing the numerator and denominator by
we have
As
the denominator approaches
As
the numerator approaches
As
the numerator approaches
Therefore
whereas
as shown in the following figure.
As
the values
As
the values
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