Limit Calculator
Both sides, kept separate, so a jump reads as a jump.
Work out Limit. Both sides, kept separate, so a jump reads as a jump. Free, with no account and nothing to install.
Try sin(x)/x at 0, or (x^2-1)/(x-1) at 1.
Limit as x → 0
1
The limit exists even though the function is undefined at the point — a removable discontinuity, or a hole.
The two one-sided limits are computed separately and only combined if they agree. That matters: a jump discontinuity has perfectly well-defined limits from each side and no limit at all, and any method that averages them reports a number the function never approaches. The approach stops at about eight orders of magnitude. Pushing closer loses more to floating-point cancellation than it gains in approach, which is how naive limit calculators start returning noise near the answer rather than the answer. This is numerical evidence, not a proof. It cannot see what a pathological function does between the sample points, and for anything unusual the algebra — factoring, L'Hôpital, a series expansion — is what actually settles it.
How the Limit Calculator works
The limit of a function at a point, computed from each side independently and only combined when they agree. A jump discontinuity has well-defined one-sided limits and no limit at all, and averaging them reports a number the function never approaches.
Also known as: limit of sin x over x · does the limit exist · one sided limit calculator · what happens as x approaches 0
Frequently asked questions
What is the limit of sin(x)/x as x approaches 0?
1. The function is undefined at 0 — it is 0/0 there — but it approaches 1 from both sides, which makes it a removable discontinuity. It is one of the foundational limits in calculus.
What is a one-sided limit?
The value a function approaches from the left or the right only. When the two differ there is no limit, which is exactly what a jump discontinuity is, and reporting a single number for that case would be wrong.
Is this a proof?
No. It is numerical evidence: the function is sampled increasingly close to the point and the values are watched. It cannot see what happens between the samples, and for anything pathological the algebra is what settles it.
Why does the approach stop short of the point?
Because floating-point cancellation takes over. Past about eight orders of magnitude the arithmetic loses more precision than the closer approach gains, which is how naive limit calculators start returning noise instead of an answer.
What is a removable discontinuity?
A point where the limit exists but the function is either undefined or takes a different value. It is removable because redefining the function at that single point makes it continuous — the hole can be filled.
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