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.
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, and the regular two-sided limit exists only when both halves agree.
In AP Calc, the left-hand limit shows up twice with two different jobs. In limits and continuity 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.
This term lives in Topic 2.4 (Connecting Differentiability and Continuity) in Unit 2, supporting learning objective 2.4.A. The essential knowledge is blunt about it: one way a continuous function 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.
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view galleryRight-hand limit (Units 1-2)
The left-hand limit's twin. The two-sided limit 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)
Piecewise functions are where one-sided limits 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 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)
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.
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.
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.
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).
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.
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.
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.
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.
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.
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