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The limit of a function

A quantity L is the limit    of a function f ( x ) as x approaches a if, as the input values of x approach a (but do not equal a ) , the corresponding output values of f ( x ) get closer to L . Note that the value of the limit is not affected by the output value of f ( x ) at a . Both a and L must be real numbers. We write it as

lim x a f ( x ) = L

Understanding the limit of a function

For the following limit, define a , f ( x ) , and L .

lim x 2 ( 3 x + 5 ) = 11

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

lim x a f ( x ) = L .

We are given

lim x 2 ( 3 x + 5 ) = 11.

This means that a = 2 , f ( x ) = 3 x + 5 ,  and  L = 11.

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For the following limit, define a , f ( x ) , and L .

lim x 5 ( 2 x 2 4 ) = 46

a = 5 , f ( x ) = 2 x 2 4 , and L = 46.

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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. [link] shows the values of

f ( x ) = x + 1 , x 7

as described earlier and depicted in [link] .

Table showing that f(x) approaches 8 from either side as x approaches 7 from either side.

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 [link] are 6.9 , 6.99 , and 6.999. The corresponding outputs are 7.9 , 7.99 , and 7.999. These values are getting closer to 8. The limit of values of f ( x ) as x approaches from the left is known as the left-hand limit. For this function, 8 is the left-hand limit of the function f ( x ) = x + 1 , x 7 as x approaches 7.

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 [link] are 7.1 , 7.01 , and 7.001. The corresponding outputs are 8.1 , 8.01 , and 8.001. These values are getting closer to 8. The limit of values of f ( x ) as x approaches from the right is known as the right-hand limit. For this function, 8 is also the right-hand limit of the function f ( x ) = x + 1 , x 7 as x approaches 7.

[link] 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 x within the interval 6.9 < x < 7.1 to produce an output value of f ( x ) within the interval 7.9 < f ( x ) < 8.1.

We also see that we can get output values of f ( x ) successively closer to 8 by selecting input values closer to 7. In fact, we can obtain output values within any specified interval if we choose appropriate input values.

[link] provides a visual representation of the left- and right-hand limits of the function. From the graph of f ( x ) , we observe the output can get infinitesimally close to L = 8 as x approaches 7 from the left and as x approaches 7 from the right.

To indicate the left-hand limit, we write

lim x 7 f ( x ) = 8.

To indicate the right-hand limit, we write

lim x 7 + f ( x ) = 8.
Graph of the previous function explaining the function's limit at (7, 8)
The left- and right-hand limits are the same for this function.

Left- and right-hand limits

The left-hand limit    of a function f ( x ) as x approaches a from the left is equal to L , denoted by

lim x a f ( x ) = L .

The values of f ( x ) can get as close to the limit L as we like by taking values of x sufficiently close to a such that x < a and x a .

The right-hand limit    of a function f ( x ) , as x approaches a from the right, is equal to L , denoted by

lim x a + f ( x ) = L .

The values of f ( x ) can get as close to the limit L as we like by taking values of x sufficiently close to a but greater than a . Both a and L are real numbers.

Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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