. Each direction gets its own answer, and those answers need not agree.

One-sided limits serve as the building blocks for two-sided limits. The two-sided limit exists precisely when both one-sided limits exist and match. When they differ, the one-sided limits capture the full story that a single two-sided limit cannot tell.

Key Terms

One-Sided Limit— limit from one direction:

\lim_{x \to a^-}

\lim_{x \to a^+}

Limit— exists when both one-sided limits agree

Discontinuity— jump discontinuities have differing one-sided limits

Continuity— one-sided continuity at interval endpoints

See All Calculus Definitions →

Right-Hand Limits

The right-hand limit uses a plus superscript:

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

This notation means

x

approaches

a

through values strictly greater than

a

. You move along the

x

-axis from the right, getting closer to

a

but never reaching or passing it.

Alternative notations include

\lim_{x \to a+} f(x)

and

\lim_{x \downarrow a} f(x)

. Some texts describe this as approaching "from above" since larger

x

-values lie above

a

on the number line.

The right-hand limit asks: as

x

decreases toward

a

, what value does

f(x)

approach?

The Connection to Two-Sided Limits

The two-sided limit exists if and only if both one-sided limits exist and are equal:

\lim_{x \to a} f(x) = L \quad \Longleftrightarrow \quad \lim_{x \to a^-} f(x) = L \;\text{ and }\; \lim_{x \to a^+} f(x) = L

One-sided limits decompose the two-sided limit into its directional components. Checking whether a two-sided limit exists often begins with computing the one-sided limits separately.

If the one-sided limits exist but differ, the two-sided limit does not exist. If either one-sided limit fails to exist (due to oscillation or unbounded behavior), the two-sided limit also fails.

Left-hand limit at a	Right-hand limit at a	Match?	Two-sided limit at a
exists, = L	exists, = L	✓	exists, equals L
exists, = L₁	exists, = L₂ ≠ L₁	✗	does not exist (jump-style failure)
exists	does not exist	—	does not exist
does not exist	exists or does not exist	—	does not exist

Reading the One-Sided Notation

Notation

Reading the One-Sided Notation

The two sections above introduced the minus and plus superscripts in passing. This section is about the marks themselves: the three competing traditions you will meet across textbooks, the shorthand for a one-sided value rather than a one-sided limit, and the two places the superscript is routinely misread.

\lim

and

\to

keep the meanings given on Limits. The criterion that both sides must agree is stated in The Connection to Two-Sided Limits above.

\lim\limits_{x \to a^{-}} f(x)

The limit of f of x as x approaches a from the left

The superscript minus is a direction, not an operation. It restricts

x

to values strictly below

a

and says nothing about the sign of anything. The function need not be defined at

a

at all — that is the whole point of a limit, and it stays true one-sided. The LaTeX reference writes it \lim_{x \to a^{-}}.

CasesAt an interior point both sides exist and the two-sided limit is the question worth asking. At the left endpoint of a domain there is nothing below

a

, so only the right-hand limit exists and continuity there is one-sided by necessity. Across a jump in a piecewise function both sides exist and disagree.

Also written

\lim\limits_{x \to a-0} f(x)

— the Russian and Eastern European tradition, standard in Fichtenholz and Demidovich, where

-0

reads as “a shade below”.

\lim\limits_{x \uparrow a} f(x)

— common in analysis and measure theory, where the arrow stresses that

x

climbs monotonically toward

a

; the mathematical keyboard carries

\uparrow

under its arrow set.

\lim\limits_{x \to a,\, x<a} f(x)

— the explicit-condition form favoured in French texts, which simply spells out what the superscript abbreviates.

Do not confuse

a^{-}

is not

-a

, and it is not

a-1

. Nothing is being subtracted. The clearest case is

x \to 0^{-}

: this means approaching zero through negative numbers, not approaching some quantity called negative zero. The superscript modifies the approach, never the number.

Same glyph elsewhereA superscript

-1

marks the inverse function — the same corner of the symbol doing an unrelated job. The one place the usage genuinely carries over is probability, where

F(x^{-})

is the left limit of a distribution function: same notation, same meaning, different subject.

\lim\limits_{x \to a^{+}} f(x)

The limit of f of x as x approaches a from the right

The mirror of the left-hand form:

x

is restricted to values strictly above

a

. Worth reading aloud as “from above” rather than “from the right”, because right and greater only coincide while you are looking at a horizontal axis.

CasesThis is the side that survives at a left endpoint — on

[a,b]

the only limit available at

a

is the right-hand one. It is also the side that defines right-continuity, the convention every cumulative distribution function obeys.

Also written

\lim\limits_{x \to a+0} f(x)

and

\lim\limits_{x \downarrow a} f(x)

, matching the two traditions above; the mathematical keyboard has

\downarrow

beside

\uparrow

Do not confuseThe arrow pairing runs opposite to intuition:

\uparrow

goes with the minus superscript and

\downarrow

goes with the plus. The arrow describes how

x

travels — climbing from below, or descending from above — while the superscript names the side it started on. Anyone who learns one form and later meets the other swaps them at least once.

f(a^{-})

f(a^{+})

f of a minus, f of a plus

Shorthand for the value of the one-sided limit, not for evaluating

f

at some point called

a^{-}

. No such point exists.

f(a^{-})

is simply a compact way of writing

\lim\limits_{x \to a^{-}} f(x)

when the limit has to appear several times in one formula.

CasesThe shorthand earns its keep wherever both sides appear together. A Fourier series converges at a jump to the midpoint

\tfrac{1}{2}\left[f(a^{-}) + f(a^{+})\right]

, which is unreadable written out in full. Right-continuity is stated as

f(a) = f(a^{+})

. The size of a jump in a piecewise function is

f(a^{+}) - f(a^{-})

Also written

f(a-0)

and

f(a+0)

, following the same tradition that writes

x \to a-0

Do not confuse

f(a)

is the actual value of the function at

a

— a third number, which may equal one side, both, or neither, and may not exist at all. A removable discontinuity is exactly the case

f(a^{-}) = f(a^{+}) \neq f(a)

, and it becomes invisible the moment the three are treated as one object.

All calculus symbols →↑ Limits

Piecewise Functions

Functions defined by different formulas on different intervals require one-sided limit analysis at the boundaries between pieces.

Consider:

f(x) = \begin{cases} x^2 & x < 2 \\ 3x - 2 & x \geq 2 \end{cases}

x = 2

, the left-hand limit uses the formula

x^2

\lim_{x \to 2^-} x^2 = 4

The right-hand limit uses the formula

3x - 2

\lim_{x \to 2^+} (3x - 2) = 4

Since both one-sided limits equal

4

, the two-sided limit exists and equals

4

. If the formulas had produced different values, the two-sided limit would not exist.

Jump Discontinuities

A jump discontinuity occurs when both one-sided limits exist but differ. The function "jumps" from one value to another at the point.

The floor function

\lfloor x \rfloor

provides a standard example. At any integer

n

\lim_{x \to n^-} \lfloor x \rfloor = n - 1

\lim_{x \to n^+} \lfloor x \rfloor = n

The left-hand limit gives the integer below, while the right-hand limit gives the integer itself. The function jumps by

1

at each integer. No two-sided limit exists at these points because the one-sided limits disagree.

One-Sided Limits at Vertical Asymptotes

Near a vertical asymptote, one-sided limits typically equal

+\infty

-\infty

. The sign can differ depending on the direction of approach.

For

f(x) = \dfrac{1}{x - 2}

\lim_{x \to 2^-} \frac{1}{x - 2} = -\infty

\lim_{x \to 2^+} \frac{1}{x - 2} = +\infty

From the left,

x - 2

is a small negative number, so the reciprocal is a large negative number. From the right,

x - 2

is a small positive number, so the reciprocal is a large positive number.

The limits and infinity page covers infinite limits in detail.

One-Sided Limits and Square Roots

Expressions involving square roots often force one-sided analysis due to domain restrictions.

The expression

\sqrt{4 - x}

requires

4 - x \geq 0

, meaning

x \leq 4

. At

x = 4

, only the left-hand limit is meaningful:

\lim_{x \to 4^-} \sqrt{4 - x} = 0

Similarly,

\sqrt{x - 4}

requires

x \geq 4

, so at

x = 4

, only the right-hand limit applies:

\lim_{x \to 4^+} \sqrt{x - 4} = 0

Domain boundaries naturally restrict limits to one side.

Evaluating One-Sided Limits

The same techniques used for two-sided limits apply: direct substitution, factoring, rationalizing. The difference lies in tracking which side you approach from.

Sign analysis becomes critical. For the expression

\dfrac{|x|}{x}

\lim_{x \to 0^+} \frac{|x|}{x} = \frac{x}{x} = 1

\lim_{x \to 0^-} \frac{|x|}{x} = \frac{-x}{x} = -1

When

x > 0

, the absolute value

|x| = x

. When

x < 0

, the absolute value

|x| = -x

. The direction of approach determines which case applies.

One-Sided Continuity

A function can be continuous from one side without being continuous from the other.

Continuous from the left at

a

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

Continuous from the right at

a

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

One-Sided Continuity

f \text{ right-continuous at } a \iff \lim_{x \to a^+} f(x) = f(a); \quad f \text{ left-continuous at } a \iff \lim_{x \to a^-} f(x) = f(a)

Learn more about this formula: One-Sided Continuity →

Browse all limit formulas →

Full continuity at

a

requires both. On a closed interval

[a, b]

, continuity means: continuous on the open interval

(a, b)

, continuous from the right at

a

, and continuous from the left at

b

Applications of One-Sided Limits

One-sided limits appear throughout calculus and its applications.

Endpoint Behavior

On closed intervals, function behavior at endpoints can only be examined from one direction. The limit from within the interval captures the boundary behavior.

Classifying Discontinuities

Determining whether a discontinuity is a jump, removable, or infinite requires comparing one-sided limits to each other and to the function value.

Piecewise Models

Real-world models often switch formulas at threshold values—tax brackets, shipping rates, material phase changes. One-sided limits detect whether the transition is smooth or abrupt.

One-Sided Derivatives

A function may have different instantaneous rates of change from the left and right at a corner point. One-sided derivatives capture this asymmetry.

Summary: Where One-Sided Limits Are Essential

One-sided limits aren't just a finer-grained version of the two-sided limit — they're the right tool for a handful of specific situations where two-sided analysis either fails or doesn't apply. The table below collects six such situations, pairing each with its diagnostic pattern: what the LHL and RHL typically look like, and why the one-sided form is structurally required. Recognizing the situation often points directly at the right technique.

Situation	Why one-sided analysis is required	Pattern of one-sided limits	Section
Piecewise function at a boundary	a different formula applies on each side of the boundary	use the left-side formula for LHL, the right-side formula for RHL; check whether they agree	obj4
Jump discontinuity	function jumps from one value to another at the point	LHL = L₁, RHL = L₂, with L₁ ≠ L₂; both finite	obj5
Vertical asymptote	function is unbounded near the point; signs may differ by direction	LHL = ±∞ and RHL = ±∞, possibly with opposite signs — see limits and infinity	obj6
Domain boundary (radicals, etc.)	function is only defined on one side of the boundary point	only one one-sided limit is meaningful; the other isn't defined	obj7
Closed-interval endpoint	continuity at a or b on [a, b] can only be tested from inside the interval	continuous-from-right at a, continuous-from-left at b	obj9
Corner point (one-sided derivatives)	instantaneous rate of change differs left vs right at a kink	left-derivative ≠ right-derivative ⇒ function is not differentiable at the point	obj10

One-Sided Limits FAQ

What does x → 0⁻ mean?

The superscript marks a direction, not an operation. Nothing is being subtracted: 0⁻ is not −0 and not 0 − 1. Writing x → 0⁻ means x approaches zero through negative values, staying strictly below it. There is no point called “negative zero” being approached — the superscript modifies the approach, never the number itself.Read more →

What does f(a⁻) mean?

It is shorthand for the value of the left-hand limit, lim(x→a⁻) f(x) — not the result of evaluating f at some point called a⁻, since no such point exists. The notation earns its keep when both sides appear together: the size of a jump is f(a⁺) − f(a⁻), and right-continuity is written f(a) = f(a⁺).Read more →

How do you find limits of piecewise functions?

At a boundary between pieces, evaluate each one-sided limit using the formula that applies on that side. Where f(x) = x² for x < 2 and f(x) = 3x − 2 for x ≥ 2, the left limit at 2 uses x² and gives 4, while the right limit uses 3x − 2 and also gives 4. Since they agree, the two-sided limit exists.Read more →

Why do square roots require one-sided limits?

Because a square root restricts the domain, and a limit can only approach through points where the function is defined. The expression √(4 − x) needs x ≤ 4, so at x = 4 nothing lies to the right and only the left-hand limit is meaningful. Mirror image for √(x − 4): only the right-hand limit exists at 4.Read more →