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 \(\lim\limits_{x\to a}f(x) = L\) 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 \(\lim\limits_{x\to 2} f(x)\) would be?

Show solution

The values \(f(x)\) get closer to \(4\) as \(x\) gets closer to \(2\). This suggests that \(\lim\limits_{x \to 2} f(x) = 4\).

Notice that \(\lim\limits_{x \to 2} f(x)\) exists even though \(f(2)\) is not defined in the table of values.

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 \(\lim\limits_{x\to 1^-} g(x)\) and \(\lim\limits_{x\to 1^+} g(x)\) would be?

Show solution

If approaching from the left (\(x< 1\)), the values \(g(x)\) get closer to \(2\) as \(x\) gets closer to \(1\). This suggests that \(\lim\limits_{x \to 1^-} g(x) = 2\).

If approaching from the right (\(x> 1\)), the values \(g(x)\) get closer to \(1\) as \(x\) gets closer to \(1\). This suggests that \(\lim\limits_{x \to 1^+} g(x) = 1\).

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

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

Example 1 versus Example 2

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

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

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