---
title: "Left-Hand Limit — AP Calc Definition & Exam Guide"
description: "A left-hand limit is the value f(x) approaches as x approaches a point from below. It's key for testing continuity, jump discontinuities, and corners like |x|."
canonical: "https://fiveable.me/ap-calc/key-terms/left-hand-limit"
type: "key-term"
subject: "AP Calculus AB/BC"
unit: "Unit 2"
---

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

## Definition

A left-hand limit, written lim x→a⁻ f(x), is the value a function approaches as x gets closer to a from values less than a. On the AP exam, you compare it to the right-hand limit to decide whether a limit exists, whether a function is continuous, and whether a derivative exists at a point.

## What It Is

A left-hand limit asks one simple question: as x slides toward a point from the left side (values smaller than a), where is f(x) heading? You write it as lim x→a⁻ f(x), with the little minus sign signaling "approach from below." It's one half of a matched pair with the [right-hand limit](/ap-calc/key-terms/right-hand-limit "fv-autolink"), and the regular two-sided limit exists only when both halves agree.

In [AP Calc](/ap-calc "fv-autolink"), the left-hand limit shows up twice with two different jobs. In [limits and continuity](/ap-calc/unit-1 "fv-autolink") work, you apply it directly to f(x), usually with piecewise functions, to check whether the two sides meet up. In Unit 2, the same idea gets applied to the *difference quotient*. If the left-hand and right-hand limits of the difference quotient disagree at a point, the derivative doesn't exist there. That's exactly what happens with f(x) = |x| at x = 0, where the left-hand limit of the difference quotient is -1 and the right-hand limit is +1. The slopes refuse to agree, so there's a corner and no derivative.

## Why It Matters

This term lives in Topic 2.4 (Connecting Differentiability and Continuity) in [Unit 2](/ap-calc/unit-2 "fv-autolink"), supporting learning objective 2.4.A. The essential knowledge is blunt about it: one way a [continuous function](/ap-calc/key-terms/continuous-function "fv-autolink") fails to be differentiable is that "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 left-hand limit is the tool you use to detect corners and cusps. It's also the foundation for continuity arguments from Unit 1, since a function is only continuous at a point if the left-hand limit, right-hand limit, and function value all match. Master this one idea and you've got the machinery behind jump discontinuities, piecewise continuity checks, and "where does f'(x) not exist" questions.

## Connections

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

The left-hand limit's twin. The [two-sided limit](/ap-calc/key-terms/two-sided-limit "fv-autolink") exists only when these two agree, and a derivative exists only when the left and right limits of the difference quotient agree. Almost every left-hand limit question is secretly a comparison question.

### [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") earn their keep. To find lim x→a⁻ f(x) at a breakpoint, you plug into the piece defined for x < a. Continuity and differentiability questions about piecewise functions are really just left-vs-right limit comparisons in disguise.

### Connecting Differentiability and Continuity (Unit 2)

Topic 2.4 upgrades the left-hand limit from checking values of f(x) to checking *slopes*. A [corner](/ap-calc/key-terms/corner "fv-autolink") like |x| at x = 0 is continuous (the function values match from both sides) but not differentiable (the slopes don't). Same tool, applied one level up.

### [Jump discontinuity (Unit 1)](/ap-calc/key-terms/jump-discontinuity)

When the left-hand and right-hand limits both exist but aren't equal, the graph literally jumps. This is the classic MCQ setup, where you identify the discontinuity type purely from one-sided limit behavior.

## On the AP Exam

Expect left-hand limits in multiple-choice stems built around piecewise functions, graphs, and tables. Common asks include evaluating lim x→a⁻ f(x) from a graph, classifying a discontinuity when the left-hand and right-hand limits disagree (that's a jump discontinuity), and finding the left-hand limit of the difference quotient for functions like f(x) = |x| at x = 0 (answer: -1, the slope from the left). On FRQs, one-sided limits typically appear inside a continuity or differentiability justification. The expected move is to compute both one-sided limits, state whether they're equal, and draw the conclusion. Writing "lim x→a⁻ f(x) = lim x→a⁺ f(x) = f(a), so f is continuous at a" is the kind of complete justification that earns the point.

## left-hand limit vs right-hand limit

The left-hand limit approaches a from values less than a (the minus sign in x→a⁻), while the right-hand limit approaches from values greater than a (x→a⁺). The classic mix-up is reading the superscript as the sign of the x-values rather than the direction of approach. The left-hand limit at x = 0 can absolutely involve negative x-values heading toward zero. Remember it as direction, not sign: minus means coming from the left on the number line.

## Key Takeaways

- The left-hand limit, lim x→a⁻ f(x), is the value f(x) approaches as x approaches a from values less than a.
- A two-sided limit exists only if the left-hand and right-hand limits exist and are equal.
- If the left-hand and right-hand limits both exist but disagree, the function has a jump discontinuity at that point.
- Per Topic 2.4, a derivative fails to exist at a corner because the left-hand and right-hand limits of the difference quotient are unequal, like f(x) = |x| at x = 0, where they equal -1 and +1.
- For a piecewise function, compute the left-hand limit at a breakpoint by using the piece defined for x-values below that point.
- A full continuity justification requires three matching things: the left-hand limit, the right-hand limit, and the actual function value f(a).

## FAQs

### What is a left-hand limit in calculus?

It's the value a function approaches as x approaches a point from values less than that point, written lim x→a⁻ f(x). It's one of the two one-sided limits you compare to check whether a full limit, continuity, or a derivative exists.

### Does the minus sign in x→a⁻ mean x is negative?

No. The superscript minus means you're approaching a from the left (from smaller values), not that x is negative. The left-hand limit as x→5⁻ uses x-values like 4.9 and 4.99, all positive.

### What's the difference between a left-hand limit and a right-hand limit?

Direction of approach. The left-hand limit (x→a⁻) comes from values below a, and the right-hand limit (x→a⁺) comes from values above a. If they're equal, the two-sided limit exists; if they both exist but differ, you have a jump discontinuity.

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

Not automatically. Equal one-sided limits mean the two-sided limit exists, but continuity also requires that limit to equal f(a). A removable discontinuity (a hole) has matching one-sided limits but still fails continuity.

### What is the left-hand limit of the difference quotient for |x| at x = 0?

It's -1, because for x < 0, the slope of |x| is -1. The right-hand limit is +1, and since the two disagree, f(x) = |x| has no derivative at x = 0. This is the CED's go-to example of a continuous but non-differentiable function in Topic 2.4.

## 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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