Limit of a Function

Limit of a Function#

Definition of a Limit#

Definition

The limit of $f (x)$ as $x$ approaches $a$ is equal to the finite number $L$ , denoted by

lim_{x \to a} f (x) = L

if the value of $f (x)$ can be made as close to $L$ as we want by taking x values sufficiently close to, but not equal to, $a$ .

The Value of a Limit versus the Value of a Function

Keep in mind that the value of a limit does not depend on the value of the function at $x = a$ . In fact, it is possible that $lim_{x \to a} f (x) = L$ even though

$f (a)$ does not exist, or
$f (a)$ exists, but is not equal to $L$ .

Example 1#

The limit of a function from a table of values

Below, you see a table in which we evaluate a function $f (x)$ for values of $x$ that are close to $2$ (but not equal to $2$ ). The first four values are for inputs $x < 2$ and the last four values are for inputs $x > 2$ .

$x$	$f (x)$
1.900000	4.63000000
1.990000	4.06030000
1.999000	4.00600300
1.999900	4.00060003
2.000100	3.99959997
2.001000	3.99599700
2.010000	3.95970000
2.100000	3.57000000

What do the values of $f (x)$ suggest $lim_{x \to 2} f (x)$ would be?

One-Sided Limits#

Definition

The limit of $f (x)$ as $x$ approaches $a$ from the right is equal to the finite number $L$ , denoted by

lim_{x \to a^{+}} f (x) = L

if the values of $f (x)$ can be made as close to $L$ as we want by taking $x$ sufficiently close to (but not equal to) $a$ and to the right of $a$ (i.e., $x > a$ ).

Similarly, the limit of $f (x)$ as $x$ approaches $a$ from the left is equal to the finite number $M$ , denoted by

lim_{x \to a^{-}} f (x) = M

if the values of $f (x)$ can be made as close to $M$ as we want by taking $x$ sufficiently close to (but not equal to) $a$ and to the left of $a$ (i.e., $x < a$ ).

Example 2#

One-sided limits of a function from a table of values

Below, you see a table in which we evaluate a function $g (x)$ for values of $x$ that are closer and closer to $1$ (but not equal to $1$ ). The first four values are for inputs $x < 1$ and the last four values are for inputs $x > 1$ .

$x$	$g (x)$
0.900000	1.810000
0.990000	1.980100
0.999000	1.998001
0.999900	1.999800
1.000100	1.000600
1.001000	1.006003
1.010000	1.060300
1.100000	1.630000

What do the values of $g (x)$ suggest $lim_{x \to 1^{-}} g (x)$ and $lim_{x \to 1^{+}} g (x)$ would be?

Limit vs. One-Sided Limits#

Connection Between Limits and One-Sided Limits

If both one-sided limits exist and are equal to the same value, $L$ , then we can say that the limit exists, and is also equal to $L$ . In other words,

lim_{x \to a} f (x) = L

means the same thing as

lim_{x \to a^{-}} f (x) = L and lim_{x \to a^{+}} f (x) = L .

Example 1 versus Example 2

In Example 1 above, we claimed that $lim_{x \to 2} f (x) = 4$ . This means that

lim_{x \to 2^{-}} f (x) = 4 and lim_{x \to 2^{+}} f (x) = 4.

In Example 2 above, we claimed that $lim_{x \to 1^{-}} g (x) = 2$ and $lim_{x \to 1^{+}} g (x) = 1$ . This means that $lim_{x \to 1} g (x)$ does not exist since the two one-sided limits exist, but are not equal to each other.