A function cannot cross a vertical asymptote because the graph must approach infinity (or
from at least one direction as
approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function
shown in
[link] intersects the horizontal asymptote
an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.
The algebraic limit laws and squeeze theorem we introduced in
Introduction to Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.
Computing limits at infinity
For each of the following functions
evaluate
and
Determine the horizontal asymptote(s) for
Using the algebraic limit laws, we have
Similarly,
Therefore,
has a horizontal asymptote of
and
approaches this horizontal asymptote as
as shown in the following graph.
Since
for all
we have
for all
Also, since
we can apply the squeeze theorem to conclude that
Similarly,
Thus,
has a horizontal asymptote of
and
approaches this horizontal asymptote as
as shown in the following graph.
To evaluate
and
we first consider the graph of
over the interval
as shown in the following graph.
Since
it follows that
Similarly, since
it follows that
As a result,
and
are horizontal asymptotes of
as shown in the following graph.
Sometimes the values of a function
become arbitrarily large as
(or as
In this case, we write
(or
On the other hand, if the values of
are negative but become arbitrarily large in magnitude as
(or as
we write
(or
For example, consider the function
As seen in
[link] and
[link] , as
the values
become arbitrarily large. Therefore,
On the other hand, as
the values of
are negative but become arbitrarily large in magnitude. Consequently,
Values of a power function as
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature