Introduction

Sometimes, a function might increase or decrease without bound when its input gets close to a specific value.

For example, consider the function whose graph is shown below:

As the input x gets closer to 1, the values of the function get larger and larger.

When this happens, we say that the function approaches "infinity", or that it has an infinite limit at x=1. We can write this mathematically, as follows:

f(x) \rightarrow \infty\quad \text{as}\quad x \rightarrow 1

In words, this reads "the function f(x) approaches infinity as x approaches 1 ".

Now, consider the function y = g(x), shown below. This is an example of a function that decreases without bound.

The function g(x) approaches negative infinity as x approaches 1. So, we can write

g(x) \rightarrow -\infty\quad \text{as } \quad x \rightarrow 1.

In words, this reads "the function g(x) approaches negative infinity as x approaches 1 ".

FLAG
Example: Identifying Points Where a Function Approaches Infinity

The graph of the function g(x) is shown above. Find the value of b-a, given the following:

  • g(x)\rightarrow -\infty as x\rightarrow a

  • g(x)\rightarrow \infty as x\rightarrow b

EXPLANATION

From the given graph, we see that as x approaches -1, the values of g(x) decrease, approaching -\infty.

So, we can write g(x) \to -\infty \quad \text{as} \quad x \to -1.

Therefore, a = -1.

From the given graph, we see that as x approaches 1, the values of g(x) increase, approaching \infty.

So, we can write g(x) \to \infty \quad \text{as} \quad x \to 1.

Therefore, b =1.

Finally,

b - a = 1 - (-1) = 2.

FLAG
Practice: Identifying Points Where a Function Approaches Infinity

The graph of the function $h(x)$ is shown above. Find the value of $a-b$ given that:

  • $h(x)\rightarrow \infty$ as $x\rightarrow a$

  • $h(x)\rightarrow -\infty$ as $x\rightarrow b$

a
 $3$
b
 $-3$
c
 $-1$
d
 $-2$
e
 $1$
Practice: Identifying Points Where a Function Approaches Infinity

The graph of $y = f (x)$ is shown above. Find the values ​​of $a$ such that $f(x) \to - \infty$ as $x \to a.$

a
 $a = -1$ and $a = 1$
b
 $a = -1$ only
c
 $a = 2$ only
d
 $a = 0$ and $a=1$
e
 $a = 1$ only
Example: Classifying Infinite Behavior of a Function

The graph of y=f(x) is shown below. Which of the following is true when x\rightarrow 2?

  1. f(x) \rightarrow \infty
  2. f(x) \rightarrow 2
  3. f(x) \rightarrow -\infty
EXPLANATION

From the given graph, we see that as x approaches 2, the values of f(x) increase, approaching \infty.

Therefore, the correct answer is "I only".

FLAG
Practice: Classifying Infinite Behavior of a Function

The graph of $y=h(x)$ is shown above. Which of the following statements are true?

  1. $h(x)\rightarrow \infty$ as $x \rightarrow 1$
  2. $h(x)\rightarrow -\infty$ as $x \rightarrow -1$
  3. $h(x)\rightarrow -\infty$ as $x \rightarrow 1$
a
 II only
b
 I and II only
c
 I only
d
 III only
e
 II and III only
Practice: Classifying Infinite Behavior of a Function

The graph of $y=f(x)$ is shown above. Which of the following is true when $x\rightarrow -1?$

a
 $f(x) \rightarrow 0$
b
 $f(x) \rightarrow 3$
c
 $f(x) \rightarrow -3$
d
 $f(x)\rightarrow \infty$
e
 $f(x)\rightarrow -\infty$
One-Sided Infinite Behavior of Functions

We know that some functions approach infinity as x approaches a certain value, like the function y=f(x) plotted below.

Here, the function shoots upwards when x approaches 2 from the right, that is, from the positive direction of the x -axis.

We say that the function approaches positive infinity as x approaches 2 from the right, and we write

f(x) \rightarrow \infty \quad \text{as} \quad x \rightarrow 2^+,

where the superscript + next to the 2 indicates that x approaches 2 from the right.

Similar to the case above, we can have a function that shoots upwards or downwards when x approaches a finite value from the left, that is, from the negative direction of the x -axis. Consider the function y=g(x), shown below.



We say that the function approaches negative infinity as x approaches 2 from the left, and we write

g(x) \rightarrow -\infty \quad \text{as} \quad x \rightarrow 2^-,

where the superscript - indicates that x approaches 2 from the left.

FLAG
Example: Recognizing One-Sided Infinite Behavior of a Function

The graph of y=f(x) is shown above. Which of the following is true when x \rightarrow 1^{+}?

  1. f(x) \rightarrow -2
  2. f(x)\rightarrow -\infty
  3. f(x)\rightarrow \infty
EXPLANATION

The notation x\rightarrow 1^{+} means that x approaches 1 from the right.

From the given graph, we see that as x approaches 1 from the right, the values of f(x) increase, approaching \infty.

Therefore, f(x)\rightarrow \infty as x \rightarrow 1^{+}.

FLAG
Practice: Recognizing One-Sided Infinite Behavior of a Function

The graph of $y=f(x)$ is shown above. Which of the following is true when $x \rightarrow -2^{-}?$

a
 $f(x) \to -1$
b
 $f(x) \to 5$
c
 $f(x) \to \infty$
d
 $f(x) \to - \infty$
e
 $f(x) \to -3$
Practice: Recognizing One-Sided Infinite Behavior of a Function

The graph of $y=f(x)$ is shown above. Which of the following is true when $x \rightarrow -3^{-}?$

a
 $f(x)\rightarrow -\infty$
b
 $f(x)\rightarrow 0$
c
 $f(x)\rightarrow 2$
d
 $f(x)\rightarrow -3$
e
 $f(x)\rightarrow \infty$
Vertical Asymptotes of Functions

We know that some functions approach infinity as x approaches a certain value from one side, like the functions y=f(x) and y=g(x) plotted below.



We call the line x=2 a vertical asymptote of the functions y = f(x) and y=g(x).

An asymptote is a line that a function gets very close to but never touches. In general, if a function f(x) has an infinite limit (positive or negative) at a point a, then we write

f(x) \rightarrow \pm \infty \quad \text{as} \quad x \rightarrow a^+

or

f(x)\rightarrow \pm\infty \quad \text{as} \quad x \rightarrow a^-

and x=a is a vertical asymptote of the function.

FLAG
Example: Identifying Vertical Asymptotes

Consider the graph of y = f(x) shown below. What is the equation of its vertical asymptote?


EXPLANATION

Recall that x = a is called a vertical asymptote of y = f(x) if f(x) \to \pm \infty as x \to a^+ or x \to a^-.

From the given graph, we see that as x approaches -1 from the left, the values of f(x) increase, approaching \infty. Also, notice that as x approaches -1 from the right, the values of f(x) decrease, approaching -\infty.

So, we can write:

  • f(x) \to \infty as x \to -1^-

  • f(x) \to -\infty as x \to -1^+

These give us the same asymptote x = -1.

FLAG
Practice: Identifying Vertical Asymptotes

Consider the graph of $y = f(x)$ shown above. What are the equations of its vertical asymptotes?

a
 $x=-2$ and $x=2$ only
b
 $x=-2$, $x=-1$, and $x=2$
c
 $x=-2$ and $x=0$ only
d
 $x = -2,$ $x=0$, and $x=2$
e
 $x=0$ and $x=2$ only
Practice: Identifying Vertical Asymptotes

Consider the graph of $y = f(x)$ shown above. What are the equations of its vertical asymptotes?

a
 The graph doesn't have vertical asymptotes
b
 $x = -1$ only
c
 $x = 1$ only
d
 $x = -2$ only
e
 $x = -2$ and $x = -1$
Flag Content
Did you notice an error, or do you simply believe that something could be improved? Please explain below.
SUBMIT
CANCEL