We are often interested in understanding how functions behave as their input becomes very large or very small. This is called the function's end behavior.
Consider the function below.
The key to understanding the end behavior is to ask the following two questions:
What is the graph doing as becomes very large? ()
What is the graph doing as becomes very small? ()
Let's now answer these questions.
First, consider On the right of the graph, as gets larger and larger, the function value "levels off" to In other words, as approaches "infinity," approaches
We can express this using mathematical notation as follows:
Now, consider On the left of the graph, as gets more and more negative, the function does not "level off." Instead, the value of grows larger and larger without bound. In other words, as approaches "negative infinity," approaches "infinity."
We can express this using mathematical notation as follows:
When determining a function's end behavior, we ignore anything happening in the middle of the graph, as this is irrelevant to a function's end behavior.
If the graph "levels off" in any of the two cases above, we've found a horizontal asymptote! In the example above, is a horizontal asymptote.
More formally, whenever approaches some finite number as or we say that is a horizontal asymptote of the function
The graph of the function is given below. Find and such that as and as
On the right of the graph, we see that as gets larger and larger, the function grows larger and larger without bound (as indicated by the arrow). We can express this as follows: Therefore,
On the left of the graph, we see that as gets more and more negative, the function levels off to the value of We can express this as follows: Therefore,
The graph of a function $y = f(x)$ is given above. Find $a$ and $b$ such that $f(x) \rightarrow a$ as $x \rightarrow \infty$ and $f(x) \rightarrow b$ as $x \rightarrow -\infty.$
|
a
|
$a=-1,\quad b=-2$ |
|
b
|
$a=-1,\quad b=2$ |
|
c
|
$a=2,\quad b=1$ |
|
d
|
$a=-2,\quad b=-1$ |
|
e
|
$a=1,\quad b=-2$ |
The graph of a function $y = f(x)$ is given above. Find the value of $a$ such that $f(x) \to a$ as $x \to \infty.$
|
a
|
$3$ |
|
b
|
$0$ |
|
c
|
$2$ |
|
d
|
$1$ |
|
e
|
$-1$ |
Which of the following statements are true for the graph shown below?
- is a horizontal asymptote of
- as
- is a horizontal asymptote of
Let's analyze each statement one-by-one.
On the right side of the graph, we see that as gets larger and larger, the function levels off to the value of Symbolically, This means that is a horizontal asymptote of Therefore, I is true.
On the left side of the graph, we see that as gets more and more negative, the function levels off to the value of Symbolically, Therefore, II is also true.
The horizontal asymptotes of are and . The line is not a horizontal asymptote because the function does not level off to on the right nor on the left. Therefore, III is false.
As a result, the correct answer is "I and II only".
Which of the following statements are true for the function $y=f(x)$ whose graph is shown above?
- $y = -1$ is a horizontal asymptote of $f(x)$
- $y = 0$ is a horizontal asymptote of $f(x)$
- $f(x) \rightarrow 2$ as $x \rightarrow \infty$
|
a
|
I, II, and III |
|
b
|
I only |
|
c
|
II and III only |
|
d
|
None of the statements are true |
|
e
|
I and II only |
What are the equations of the horizontal asymptotes of $y = f(x)$ whose graph is shown above?
|
a
|
$x = -4$ and $x = 2$ |
|
b
|
$x = -2$ and $x = 1$ |
|
c
|
$y = -4$ and $y = 2$ |
|
d
|
The function has no horizontal asymptotes |
|
e
|
$y = -2$ and $y = 1$ |