What is a left-hand limit?

A left-hand limit, written lim(x→a⁻) f(x), examines function behavior as x approaches a through values strictly less than a. You move along the x-axis from the left, getting closer to a but never reaching it.

How do one-sided limits relate to two-sided limits?

The two-sided limit exists if and only if both one-sided limits exist and are equal. If the left-hand and right-hand limits differ, or if either fails to exist, the two-sided limit does not exist.

How do you find limits of piecewise functions?

At boundaries between pieces, evaluate each one-sided limit using the formula that applies on that side. For example, at x = 2 where f(x) = x² for x < 2 and f(x) = 3x−2 for x ≥ 2, use x² for the left limit and 3x−2 for the right limit.

What is a jump discontinuity?

A jump discontinuity occurs when both one-sided limits exist but differ. The function 'jumps' from one value to another at that point. The floor function ⌊x⌋ has jump discontinuities at every integer, jumping by 1 each time.

How do one-sided limits behave at vertical asymptotes?

Near vertical asymptotes, one-sided limits typically equal +∞ or −∞, and the sign can differ by direction. For f(x) = 1/(x−2), the left limit at x = 2 is −∞ while the right limit is +∞.

Why do square roots require one-sided limits?

Square root expressions have domain restrictions. For √(4−x), the domain requires x ≤ 4, so at x = 4 only the left-hand limit is meaningful. Domain boundaries naturally restrict limits to one side.

How do you evaluate one-sided limits?

Use the same techniques as two-sided limits—direct substitution, factoring, rationalizing—but track which side you approach from. Sign analysis is critical: for |x|/x, the left limit at 0 is −1 while the right limit is +1.

What is one-sided continuity?

A function is continuous from the left at a if lim(x→a⁻) f(x) = f(a), and continuous from the right if lim(x→a⁺) f(x) = f(a). Full continuity requires both. On closed intervals, endpoints use one-sided continuity.

One-Sided Limits

Q: What is a right-hand limit?

A right-hand limit, written lim(x→a⁺) f(x), examines function behavior as 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 it.

Q: What is a jump discontinuity?

A jump discontinuity occurs when both one-sided limits exist but differ. The function 'jumps' from one value to another at that point. The floor function ⌊x⌋ has jump discontinuities at every integer, jumping by 1 each time.

Q: How do one-sided limits behave at vertical asymptotes?

Near vertical asymptotes, one-sided limits typically equal +∞ or −∞, and the sign can differ by direction. For f(x) = 1/(x−2), the left limit at x = 2 is −∞ while the right limit is +∞.

Q: Why do square roots require one-sided limits?

Square root expressions have domain restrictions. For √(4−x), the domain requires x ≤ 4, so at x = 4 only the left-hand limit is meaningful. Domain boundaries naturally restrict limits to one side.

Q: How do you evaluate one-sided limits?

Use the same techniques as two-sided limits—direct substitution, factoring, rationalizing—but track which side you approach from. Sign analysis is critical: for |x|/x, the left limit at 0 is −1 while the right limit is +1.

Q: What is one-sided continuity?

A function is continuous from the left at a if lim(x→a⁻) f(x) = f(a), and continuous from the right if lim(x→a⁺) f(x) = f(a). Full continuity requires both. On closed intervals, endpoints use one-sided continuity.

Left-Hand Limits

Right-Hand Limits

The Connection to Two-Sided Limits

Piecewise Functions

Jump Discontinuities

One-Sided Limits at Vertical Asymptotes

One-Sided Limits and Square Roots

Evaluating One-Sided Limits

One-Sided Continuity

Applications of One-Sided Limits

Summary: Where One-Sided Limits Are Essential

Approaching From One Direction

Sometimes a function behaves differently depending on which side you approach from. A piecewise function may follow one formula for

x < a

and a different formula for

x > a

. A rational function may blow up to

+\infty

on one side and

-\infty

on the other. In these situations, one-sided limits become essential.

The left-hand limit examines behavior as

x

approaches

a

through values less than

a

. The right-hand limit examines behavior through values greater than

a

. 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

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)

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