One-Sided Limits

Sometimes a function approaches different values depending on the direction from which $x$ approaches a point. One-sided limits capture exactly this directional behavior, and the two-sided limit exists precisely when both one-sided limits agree.

Concept	Quick Reference
Left-hand limit	$lim_{x \to a^{-}} f (x) = L$ : approach from the left ( $x < a$ )
Right-hand limit	$lim_{x \to a^{+}} f (x) = L$ : approach from the right ( $x > a$ )
Two-sided limit exists	if and only if left and right limits both equal $L$
Alternate notation	$f (a^{-})$ for left-hand limit; $f (a^{+})$ for right-hand limit

Approaching from the Left

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 $5$. We say:

"The limit of $F (x)$ as $x$ approaches $2.5$ from the left is $5$,"

or equivalently, "the left-hand limit of $F (x)$ as $x$ approaches $2.5$ is $5$." The notation is:

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

The minus sign written after $2.5$ means $x$ approaches $2.5$ from the left (i.e., through values less than $2.5$).

Graph of y=F(x) illustrating left-hand limit of 5 and right-hand limit of 2 at x=2.5

Approaching from the Right

Now consider $x$ approaching $2.5$ from the right (through values larger than $2.5$). As $x \to {2.5}^{+}$ , $F (x)$ approaches $2$. We write:

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

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

Note that the value of $F (2.5)$ has no bearing on either one-sided limit. Even if $F$ were not defined at $x = 2.5$ , the one-sided limits would remain unchanged.

Formal Definitions

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 $L$ " and write

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

This limit is sometimes denoted $f (a^{-})$ .

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 $L$ " and write

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

This limit is sometimes denoted $f (a^{+})$ .

Connection Between One-Sided and Two-Sided Limits

$lim_{x \to a} f (x)$ exists and equals $L$ if and only if both one-sided limits exist and equal $L$ :

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

It follows that if $lim_{x \to a^{-}} f (x) \neq lim_{x \to a^{+}} f (x)$ , then the two-sided limit $lim_{x \to a} f (x)$ does not exist.

Returning to the example above: since $lim_{x \to {2.5}^{-}} F (x) = 5 \neq 2 = lim_{x \to {2.5}^{+}} F (x)$ , the limit $lim_{x \to 2.5} F (x)$ does not exist.

Frequently Asked Questions

What does the superscript

-

+

mean in the limit notation?

The superscript indicates the direction of approach.

x \to a^{-}

means

x

approaches

a

through values *less than*

a

(from the left).

x \to a^{+}

means

x

approaches

a

through values *greater than*

a

(from the right). Neither one requires

x

to equal

a

If both one-sided limits exist but are different, what can I say about the two-sided limit?

The two-sided limit does not exist. The theorem says the two-sided limit exists *if and only if* both one-sided limits exist *and are equal*. If they differ, there is no single number that

f (x)

approaches from both sides simultaneously.

Does the function value

f (a)

affect the one-sided limits?

No. One-sided limits depend only on the behavior of

f

for

x

near

a

(from one direction), not on the value of

f

a

itself. The function need not even be defined at

a

for the one-sided limits to exist.

Approaching from the Left

Approaching from the Right

Formal Definitions

Connection Between One-Sided and Two-Sided Limits

Frequently Asked Questions

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