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 graph of
crosses its horizontal asymptote
an infinite number of times.
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.
This function approaches a horizontal asymptote as
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.
This function crosses its horizontal asymptote multiple times.
To evaluate
and
we first consider the graph of
over the interval
as shown in the following graph.
The graph of
has vertical asymptotes at
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
For this function, the functional values approach infinity as
Questions & Answers
I'm interested in biological psychology and cognitive psychology
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
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miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
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Samuel
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