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
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] .
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.
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.
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.
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.