Limit from the Right — Definition, Formula & Examples

Q: What is the difference between a limit from the right and a limit from the left?

A limit from the right ($\lim_{x \to c^+}$) only considers values of $x$ that are greater than $c$, while a limit from the left ($\lim_{x \to c^-}$) only considers values less than $c$. If both one-sided limits exist and are equal, the two-sided limit $\lim_{x \to c} f(x)$ exists and equals that common value. If they differ, the two-sided limit does not exist.

Q: What does the plus sign mean in $\lim_{x \to c^+}$?

The superscript plus sign ($+$) indicates the direction of approach. It means $x$ approaches $c$ from values greater than $c$—that is, from the right side on a number line. This notation is sometimes also written as $\lim_{x \downarrow c}$ or $\lim_{x \to c+}$ without the superscript, though the $c^+$ form is most common in textbooks.

Limit from the Right
Limit from Above

A one-sided limit which, in the example $\displaystyle \lim_{x \to 0^+}\frac{1}{x} = \infty\text{,}$ restricts x such that x > 0.

In general, a limit from the right restricts domain variable to values greater than the number the domain variable approaches. When a limit is taken from the right it is written $\displaystyle \lim_{x \to a^+} f(x)$ or $\displaystyle \lim_{x \downarrow a} f(x)\text{.}$

For example, $\displaystyle \lim_{x \to 0^+}\frac{1}{x} = \infty$ since $\displaystyle \frac{1}{x}$ tends toward ∞ as x gets closer and closer to 0 from the right.

Formal Definitions:

1. $\displaystyle \lim_{x\to a^+} f(x) = L$ if and only if for each $\varepsilon \gt 0$ there exists a $\delta \gt 0$ such that
$a \lt x \lt a + \delta \Rightarrow |f(x)-L| \lt \varepsilon\text{.}$

2. $\displaystyle \lim_{x\to a^+} f(x) = \infty$ if and only if for each $N$ there exists a $\delta \gt 0$ such that
$a \lt x \lt a + \delta \Rightarrow f(x) \gt N\text{.}$

3. $\displaystyle \lim_{x\to a^+} f(x) = -\infty$ if and only if for each $N$ there exists a $\delta \gt 0$ such that
$a \lt x \lt a + \delta \Rightarrow f(x) \lt N\text{.}$

See also

Limit from the left, infinity

Key Formula

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

Where:

$x$ = The input (domain) variable approaching the value c
$c$ = The number that x approaches, from values greater than c
$f(x)$ = The function being evaluated
$L$ = The value the function approaches (the limit)
$+$ = The superscript plus sign indicates the approach is from the right (values greater than c)

Worked Example

Problem: Find the limit from the right:

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

Step 1: Identify the point of approach and the direction. Here, x approaches 0 from the right, meaning x takes positive values close to 0.

x \to 0^+ \quad \text{means} \quad x = 0.1,\; 0.01,\; 0.001, \ldots

Step 2: Evaluate the function at values approaching 0 from the right.

f(0.1) = \frac{1}{0.1} = 10, \quad f(0.01) = 100, \quad f(0.001) = 1000

Step 3: Observe the trend. As x gets closer to 0 from the right, the output grows without bound.

\frac{1}{x} \to +\infty \quad \text{as} \quad x \to 0^+

Step 4: State the result. The function increases without limit, so we say the right-hand limit is positive infinity.

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

Answer:

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

. The function grows without bound as

x

approaches 0 from the right.

Another Example

This example shows a case where the function is only defined on one side of the point. A limit from the left does not exist here because $\sqrt{x-2}$ is undefined for $x < 2$ . The right-hand limit is the only one-sided limit that makes sense at this endpoint.

Problem: Find the limit from the right:

\lim_{x \to 2^+} \sqrt{x - 2}

Step 1: Identify the point and direction. We approach x = 2 from values greater than 2, so x > 2.

x \to 2^+ \quad \text{means} \quad x = 2.1,\; 2.01,\; 2.001, \ldots

Step 2: Check the domain. The square root function

\sqrt{x-2}

is only defined when

x - 2 \geq 0

, i.e.,

x \geq 2

. So the function only exists on the right side of 2.

\text{Domain: } x \geq 2

Step 3: Evaluate the function at values approaching 2 from the right.

f(2.1) = \sqrt{0.1} \approx 0.316, \quad f(2.01) = \sqrt{0.01} = 0.1, \quad f(2.001) = \sqrt{0.001} \approx 0.0316

Step 4: As x approaches 2 from the right, the expression inside the square root approaches 0, so the output approaches 0.

\lim_{x \to 2^+} \sqrt{x - 2} = \sqrt{0} = 0

Answer:

\lim_{x \to 2^+} \sqrt{x - 2} = 0

Frequently Asked Questions

What is the difference between a limit from the right and a limit from the left?

A limit from the right (

\lim_{x \to c^+}

) only considers values of

x

that are greater than

c

, while a limit from the left (

\lim_{x \to c^-}

) only considers values less than

c

. If both one-sided limits exist and are equal, the two-sided limit

\lim_{x \to c} f(x)

exists and equals that common value. If they differ, the two-sided limit does not exist.

When do you need to use the limit from the right?

You use the limit from the right when a function behaves differently on either side of a point, such as piecewise functions or functions with vertical asymptotes. It is also essential when the function is only defined for values greater than or equal to the point, like

\sqrt{x}

x = 0

. Right-hand limits appear frequently when analyzing continuity and when determining whether a two-sided limit exists.

What does the plus sign mean in

\lim_{x \to c^+}

The superscript plus sign (

+

) indicates the direction of approach. It means

x

approaches

c

from values greater than

c

—that is, from the right side on a number line. This notation is sometimes also written as

\lim_{x \downarrow c}

\lim_{x \to c+}

without the superscript, though the

c^+

form is most common in textbooks.

Limit from the Right vs. Limit from the Left

	Limit from the Right	Limit from the Left
Notation	$\lim_{x \to c^+} f(x)$	$\lim_{x \to c^-} f(x)$
Direction of approach	x approaches c from values greater than c	x approaches c from values less than c
Restriction on x	x > c	x < c
Also called	Right-hand limit, limit from above	Left-hand limit, limit from below
Example: $\lim 1/x$ as $x \to 0$	$+\infty$	$-\infty$

Why It Matters

Right-hand limits appear constantly in calculus courses when you study continuity, derivatives, and the behavior of functions near discontinuities or domain boundaries. For piecewise-defined functions, you must check both one-sided limits to determine whether the overall limit exists at a boundary point. Understanding right-hand limits is also critical for analyzing vertical asymptotes, improper integrals, and the convergence of sequences and series.

Common Mistakes

Mistake: Confusing the direction: thinking

x \to c^+

means x approaches from the left (smaller values).

Correction: The plus sign means x approaches c from values greater than c (the right side on a number line). Think of it as x = c + a tiny positive amount that shrinks to zero.

Mistake: Assuming that if the right-hand limit exists, the two-sided limit must also exist.

Correction: The two-sided limit exists only if both the right-hand and left-hand limits exist and are equal. For example,

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

and

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

, so

\lim_{x \to 0} \frac{1}{x}

does not exist.

Related Terms

One-Sided Limit — General category that includes right-hand limits
Limit from the Left — The other one-sided limit, approaching from below
Domain — Right-hand limits restrict the domain to x > c
Variable — The input variable x whose approach direction is restricted
Infinity — Right-hand limits can equal positive or negative infinity
Continuous — Continuity requires one-sided limits to agree
Piecewise Function — Often requires separate one-sided limits at boundaries