We have seen how a sequence can have a limit, a value that the sequence of terms moves toward as the nu mber of terms increases. For example, the terms of the sequence

gets closer and closer to 0. A sequence is one type of function, but functions that are not sequences can also have limits. We can describe the behavior of the function as the input values get close to a specific value. If the limit of a function*y*-coordinate gets closer and closer to

Figure 1 provides a visual representation of the mathematical concept of limit. As the input value

We write the equation of a limit as

This notation indicates that as

Consider the function

We can factor the function as shown.

Notice that

What happens at

indicates that as the input

What happens at

This notation indicates that 7 is not in the domain of the function. We had already indicated this when we wrote the function as

Notice that the limit of a function can exist even when

### A General Note: **The Limit of a Function: **

A quantity

### Example 1

#### Problem 1

**Understanding the Limit of a Function**

For the following limit, define

##### Solution

First, we recognize the notation of a limit. If the limit exists, as

We are given

This means that

### Try It:

#### Exercise 1

For the following limit, define

##### Solution

**Understanding Left-Hand Limits and Right-Hand Limits**

We can approach the input of a function from either side of a value—from the left or the right. Figure 3 shows the values of

as described earlier and depicted in Figure 2.

Values described as “from the left” are less than the input value 7 and would therefore appear to the left of the value on a number line. The input values that approach 7 from the left in Figure 3 are

Values described as “from the right” are greater than the input value 7 and would therefore appear to the right of the value on a number line. The input values that approach 7 from the right in Figure 3 are

Figure 3 shows that we can get the output of the function within a distance of 0.1 from 8 by using an input within a distance of 0.1 from 7. In other words, we need an input

We also see that we can get output values of

Figure 4 provides a visual representation of the left- and right-hand limits of the function. From the graph of

To indicate the left-hand limit, we write

To indicate the right-hand limit, we write

#### A General Note: **Left- and Right-Hand Limits: **

The left-hand limit of a function

The values of

The right-hand limit of a function

The values of

**Understanding Two-Sided Limits**

In the previous example, the left-hand limit and right-hand limit as

#### A General Note: **The Two-Sided Limit of Function as ***x* Approaches *a* :

*x*Approaches

*a*:

The limit of a function

if and only if

In other words, the left-hand limit of a function

AnalysisRecall that
y = 3 x + 5
y = 3 x + 5
is a line with no breaks. As the input values approach 2, the output values will get close to 11. This may be phrased with the equation
lim
x → 2
( 3 x + 5 ) = 11
,
lim
x → 2
( 3 x + 5 ) = 11
, which means that as
x
x nears 2 (but is not exactly 2), the output of the function
f ( x ) = 3 x + 5
f ( x ) = 3 x + 5
gets as close as we want to
3 ( 2 ) + 5 ,
3 ( 2 ) + 5 ,
or 11, which is the limit
L ,
L ,
as we take values of
x
x sufficiently near 2 but not at
x = 2.
x = 2.