One Sided Limits

In this section, we study one-sided limits (left-hand limits and right-hand limits) which examine the behavior of a function as the input approaches a point from a single direction.

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Limit Type	Symbolic Notation	Alternative Notation	Condition
Left-Hand Limit	$lim_{x \to a^{-}} f (x) = L$	$f (a^{-}) = L$	x approaches a through values where $x < a$ .
Right-Hand Limit	$lim_{x \to a^{+}} f (x) = L$	$f (a^{+}) = L$	x approaches a through values where $x > a$ .
Two-Sided Limit Existence	$lim_{x \to a} f (x) = L$	$lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = L$	Two-sided limit exists if and only if both one-sided limits exist and are equal.

Consider the function $F (x)$ , whose graph is shown below. If we take x values closer and closer to 2.5, but less than 2.5, $F (x)$ gets closer and closer to 5. In other words, when x approaches 2.5 through the values less than 2.5, $F (x)$ approaches 5. We express this by saying that "the limit of $F (x)$ as x approaches 2.5 from the left is 5" or "the left-hand limit of $F (x)$ as x approaches 2.5 is 5." The notation for this is

lim_{x \to {2.5}^{-}} F (x) = 5.

The minus sign that is written after 2.5 means x approaches 2.5 from the left.

Graph of y = F(x) showing left and right hand limits at x = 2.5 — Graph of $y = F (x)$ .

Now consider the case in which x takes on the values close to 2.5 but larger than 2.5. As x approaches 2.5 from the right, $F (x)$ approaches 2. Symbolically we write

lim_{x \to {2.5}^{+}} F (x) = 2,

and say "the limit of $F (x)$ as x approaches 2.5 from the right is 2" or "the right-hand limit of $F (x)$ as x approaches 2.5 is 2."

In this example, $F (x)$ is defined at $x = 2.5$ , but the value of $F (2.5)$ has no bearing on the left-hand or right-hand limit of $F (x)$ . Even if we remove $x = 2.5$ from the domain of $F (x)$ (that is, if $F (x)$ were not defined at $x = 2.5$ ), the left-hand and right-hand limits will remain the same.

Intuitive Definition of Left-Hand Limit: If we can make the values of $f (x)$ as close as we please to a number $L$ by taking x sufficiently close to a with $𝒙 < 𝒂$ , we say "the limit of $f (x)$ as x approaches a from the left is $L$ " or "the left-hand limit of $f (x)$ as x approaches a is $L$ " and write

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

The above limit is sometimes denoted by $f (a^{-})$ .

Similarly:

Intuitive Definition of Right-Hand Limit: If we can make the values of $f (x)$ as close as we please to a number $L$ by taking x sufficiently close to a with $𝒙 > 𝒂$ , we say "the limit of $f (x)$ as x approaches a from the right is $L$ " or "the right-hand limit of $f (x)$ as x approaches a is $L$ " and write

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

The above limit is sometimes denoted by $f (a^{+})$ .

By comparing the definitions of one-sided limits and two-sided (or regular) limits, we realize the following is true.

$lim_{x \to a} f (x)$ exists and is equal to $L$ if and only if $lim_{x \to a^{-}} f (x)$ and $lim_{x \to a^{+}} f (x)$ both exist and are equal to $L$ . That is,

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

It follows from the above theorem that if $lim_{x \to a^{-}} f (x) \neq lim_{x \to a^{+}} f (x)$ , then $lim_{x \to a} f (x)$ does not exist. For instance, going back to the above figure, $lim_{x \to 2.5} F (x)$ does not exist because the left and the right limits of F as x approaches 2.5 are not equal.

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