What it shows
The limit of f(x) as x approaches a is the number f(x) gets close to when x gets close to a, from both sides, whatever happens at a itself. Often f(a) is fine and the limit is just f(a). The interesting cases are the ones where you cannot simply put x = a in: 0/0 forms like sin x / x at 0, steps, and curves that shoot off to infinity.
This page draws the function around the point and lets you zoom in (down to a window of ± 0.0004). Open circles show where the graph heads from each side and a filled dot shows the value f(a), if there is one. Two tables list f(x) at x = a − 0.1, a − 0.01, … and a + 0.1, a + 0.01, … so you can watch the numbers settle (or not).
It follows limits and derivatives in Class 11 maths and continuity and differentiability in Class 12 maths. Limits are what the derivative slope page is built on; the unit circle page explains radians, which the sin x / x limit needs; and the graphing calculator draws any curve you want to explore further.
How to use it
- Pick an example, or choose My own…, type f(x) (such as
(x^3 - 1)/(x - 1)) and the point a. - Slide Zoom or press Zoom in to close in on the point. Zoom out starts again.
- Read the left-hand and right-hand limits. If they agree, the limit exists. If they differ, or one runs off to infinity, or the values never settle, there is no limit.
- Switch the Level to Class 12 to see whether f is continuous at a, and what kind of break it has if not.
- For sin x / x, a second picture shows the sector argument: slide the angle and read the three areas.
The kinds of point
- Continuous: f(a) is defined, the limit exists, and they are equal. A corner (like |x| at 0) is still continuous.
- Removable break (a hole): the limit exists but f(a) is missing or different. Defining f(a) as the limit mends it.
- Jump: the left and right limits are different numbers, like |x| / x or the greatest integer function [x].
- Infinite break: the values grow without bound, like 1/x² at 0.
- No limit (wobbles): the values keep moving, like sin(1/x) at 0.
Worked example
For (x² − 4)/(x − 2) near 2 the formula gives 0/0 at x = 2. The table from the left reads 3.9, 3.99, 3.999, … and from the right 4.1, 4.01, 4.001, … so the limit is 4. Factoring shows why: (x − 2)(x + 2)/(x − 2) = x + 2 whenever x ≠ 2, a straight line with a hole at (2, 4).
For sin x / x, at x = 0.1 the value is 0.998334 and at 0.01 it is 0.999983: the limit is 1. The sector picture gives the reason: comparing areas, cos x < sin x / x < 1, and cos x → 1.
How the page finds a limit
It works out f at 0.1, 0.01, … 0.000001 either side of a. If the values settle, it reports the limit (sharpened with one extrapolation step); if they grow by a large factor each step it reports ±∞; otherwise it says there is no limit. The examples were checked against exact answers from sympy. For your own functions this is a numerical estimate, which can be fooled by very strange functions, so treat it as a guide and confirm with algebra.
Check yourself
Five fresh questions: a factorising limit, sin(kx)/x, the standard limit (xⁿ − aⁿ)/(x − a), a one-sided limit of [x], and the kind of break of 1/(x − p)². Nothing is saved.
FAQ
Why can a limit exist when f(a) does not? The limit only looks at values near a, never at a itself.
Is this from my textbook? It covers the standard school topics with its own examples and pictures. It is an independent study aid.
Where the numbers come from
- Standard calculus: one-sided limits, the limit sin x / x → 1 by comparing areas, the standard limits (xⁿ − aⁿ)/(x − a) → n aⁿ⁻¹ and (eˣ − 1)/x → 1, and the definition of continuity. Written for fibub; no textbook text, figures or exercises are used.
- Checked against sympy 1.14: left-hand and right-hand limits of every example and of three typed functions (sympy.limit with '-' and '+').
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.