---
title: "Right-Hand Limit — AP Calc Definition & Exam Guide"
description: "A right-hand limit is the value f(x) approaches as x approaches a from values greater than a. It's key for continuity checks and spotting where derivatives fail."
canonical: "https://fiveable.me/ap-calc/key-terms/right-hand-limit"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 2"
---

# Right-Hand Limit — AP Calc Definition & Exam Guide

## Definition

A right-hand limit, written lim x→a⁺ f(x), is the value a function approaches as x approaches a from values greater than a (from the right side). On the AP exam, you compare it with the left-hand limit to test continuity and, in Topic 2.4, to decide whether a derivative exists.

## What It Is

A right-hand limit asks one specific question: as x slides toward a from the right side of the number line (values bigger than a), what y-value is f(x) heading toward? The notation is lim x→a⁺ f(x), and that little plus sign means "approaching from values greater than a," not "positive numbers."

The [two-sided limit](/ap-calc/key-terms/two-sided-limit "fv-autolink") exists only when the right-hand limit and the left-hand limit agree. That single idea powers two big AP skills. First, checking [continuity](/ap-calc/unit-1/exploring-types-discontinuities/study-guide/w0TgEsiaFrXpMnMus4he "fv-autolink"), especially for piecewise functions at the point where the rule changes. Second, checking differentiability in Topic 2.4. There, you apply one-sided limits not to f(x) itself but to the **difference quotient**. For f(x) = |x| at x = 0, the right-hand limit of the difference quotient is 1 and the left-hand limit is -1. They disagree, so f'(0) doesn't exist even though the function is perfectly continuous there. That's the corner.

## Why It Matters

This term lives in [Unit 2](/ap-calc/unit-2 "fv-autolink") (Differentiation: Definition and Fundamental Properties), specifically Topic 2.4, and supports learning objective 2.4.A: explain the relationship between differentiability and continuity. The essential knowledge spells out exactly where right-hand limits earn their keep. A [continuous function](/ap-calc/key-terms/continuous-function "fv-autolink") can still fail to be differentiable, and one way that happens is when the left-hand and right-hand limits of the difference quotient are not equal, as in f(x) = |x| at x = 0. In other words, the right-hand limit is the tool that lets you prove a corner exists. If you can compute one-sided limits of a difference quotient and compare them, you can explain why differentiability is a stricter condition than continuity, which is the whole point of Topic 2.4.

## Connections

### [Left-hand limit (Units 1-2)](/ap-calc/key-terms/left-hand-limit)

These are two halves of one idea. The two-sided limit exists only when the left-hand and right-hand limits match, and a derivative exists only when the left-hand and right-hand limits of the [difference quotient](/ap-calc/key-terms/difference-quotient "fv-autolink") match. Almost every right-hand limit problem on the exam is secretly a comparison problem.

### [Piecewise function (Unit 1)](/ap-calc/key-terms/piecewise-function)

Piecewise functions are where [one-sided limits](/ap-calc/key-terms/one-sided-limits "fv-autolink") actually get computed. At the breakpoint, you plug into the right-side rule for the right-hand limit and the left-side rule for the left-hand limit. If the pieces disagree, the function jumps; if the slopes disagree, the derivative dies.

### [Closed Interval (Unit 1)](/ap-calc/key-terms/closed-interval)

At the left endpoint a of a [closed interval](/ap-calc/key-terms/closed-interval "fv-autolink") [a, b], there's nothing to the left of a inside the domain, so continuity there is defined using only the right-hand limit. Endpoints are the one place where a one-sided limit does the whole job by itself.

### Differentiability and continuity (Unit 2)

Topic 2.4's headline result, [differentiable](/ap-calc/key-terms/differentiable "fv-autolink") implies continuous but not the other way around, is proven by counterexample using right-hand limits. f(x) = |x| at x = 0 is continuous, but the right-hand limit of its difference quotient is 1 while the left-hand limit is -1, so no derivative exists.

## On the AP Exam

Right-hand limits show up two ways. In limits-and-continuity multiple choice, you'll read lim x→a⁺ off a graph or compute it from the correct piece of a piecewise function. In Unit 2, the stem changes. Practice questions ask things like "For f(x) = |x|, what is the right-hand limit of the difference quotient at x = 0?" (answer: 1) or "why does the derivative not exist at x = 0?" The move you must make is the same every time. Compute the right-hand limit, compute the left-hand limit, compare. Equal means the limit (or derivative) exists; unequal means it doesn't, and you should say why in your justification. Absolute value functions like f(x) = |x - 3| + 2 are the classic trap, continuous everywhere but not differentiable at the corner, and one-sided limits of the difference quotient are how you prove it.

## right-hand limit vs Left-hand limit

The superscript trips people up. lim x→a⁺ means x approaches a from values greater than a (from the right on the number line), and lim x→a⁻ means from values less than a. The + and - describe which side of a you're coming from, not whether x is positive or negative. So lim x→-2⁺ still means approaching -2 from the right, through values like -1.9 and -1.99.

## Key Takeaways

- A right-hand limit, lim x→a⁺ f(x), is the value f(x) approaches as x approaches a from values greater than a.
- The plus sign in x→a⁺ means "from the right side of a," not "positive x-values."
- A two-sided limit exists only if the right-hand limit and left-hand limit are equal.
- In Topic 2.4, you apply right-hand limits to the difference quotient. If the right-hand and left-hand limits of the difference quotient disagree, the derivative does not exist at that point.
- For f(x) = |x| at x = 0, the right-hand limit of the difference quotient is 1 and the left-hand limit is -1, which is exactly why the corner has no derivative.
- For piecewise functions, compute the right-hand limit using the rule that applies for x-values greater than the breakpoint.

## FAQs

### What is a right-hand limit in AP Calc?

It's the value a function approaches as x approaches a point from values greater than that point, written lim x→a⁺ f(x). You compare it with the left-hand limit to decide whether the full limit exists.

### Does x→a⁺ mean x is positive?

No. The plus sign means you're approaching a from the right side, through values greater than a. For example, lim x→-2⁺ means approaching -2 through values like -1.9 and -1.99, all of which are negative.

### What's the difference between a right-hand limit and a regular (two-sided) limit?

A two-sided limit requires the function to approach the same value from both directions. The right-hand limit only checks the approach from values greater than a. The two-sided limit exists if and only if both one-sided limits exist and are equal.

### If the right-hand and left-hand limits are equal, is the function differentiable there?

Not necessarily. Equal one-sided limits of f(x) give you the limit (and possibly continuity), but differentiability needs the one-sided limits of the difference quotient to match too. f(x) = |x| at x = 0 has matching limits of the function but mismatched difference-quotient limits (1 from the right, -1 from the left), so it's continuous but not differentiable.

### What is the right-hand limit of the difference quotient for f(x) = |x| at x = 0?

It's 1, because for x > 0 the difference quotient is |x|/x = 1. The left-hand limit is -1, and since 1 ≠ -1, f'(0) does not exist. This is the standard Topic 2.4 example of continuity without differentiability.

## Related Study Guides

- [2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist](/ap-calc/unit-2/determining-when-derivatives-do-do-not-exist/study-guide/Lk6aXtIExtqciduDNGhk)

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