How to use the Limit Calculator
- Choose the function to examine.
- Enter the x value being approached — often a point where the function is undefined.
- Compare the left and right limits: they must agree for a two-sided limit to exist.
- Read the two-sided limit, or the explicit statement that it does not exist.
- Compare against the value at x itself; a mismatch means the function is discontinuous there.
How the calculation works
A limit describes where a function is heading as the input approaches a value, independent of what happens exactly at that value. The numeric approach is direct: evaluate the function very slightly below the target and very slightly above it. If both evaluations converge on the same number, that is the limit. The distinction between the limit and the function's value at the point is precisely what makes limits useful — removable discontinuities, derivatives and asymptotes all live in that gap.
One-sided behaviour carries real information. The reciprocal function near zero dives to negative infinity from the left and climbs to positive infinity from the right, so the two-sided limit does not exist even though each side is perfectly well behaved. A step function jumps between two finite values with no agreement. Reporting the sides separately makes the failure mode visible instead of hiding it behind a single misleading number.
The numeric method has a known weakness: it samples at a fixed small offset, so it detects broad trends but can be fooled by pathological functions that oscillate infinitely fast near the point, such as sin(1/x) as x approaches zero. It also inherits floating-point cancellation for indeterminate forms like 0/0, where the true limit may be finite but the arithmetic is unstable. For those cases, algebraic simplification or L'Hôpital's rule remains the reliable route.
Left: lim(x→a⁻) f(x) ≈ f(a − ε); Right: lim(x→a⁺) f(x) ≈ f(a + ε); the two-sided limit exists only if they agree, with ε ≈ 10⁻⁷Source: Epsilon-delta definition of a limit; see OpenStax Calculus Volume 1, Chapter 2.
Worked example
Examine 1/x as x approaches zero — the standard illustration of a limit that fails to exist.
- Select 1/x and set the approach point to 0.
- From the left: 1/(−0.0000001) = −10,000,000.
- From the right: 1/(0.0000001) = +10,000,000.
- The two sides diverge in opposite directions.
No two-sided limit exists. The function has a vertical asymptote at x = 0, unbounded below on the left and above on the right.
Frequently asked questions
Can a limit exist where the function is undefined?+
Yes, and that is the most useful case. (x² − 1)/(x − 1) is undefined at x = 1 but has a limit of 2 there, which is exactly what a removable discontinuity means.
Why do the two sides sometimes differ by a tiny amount?+
Round-off. The calculator treats agreement within a small tolerance as convergence; differences below that threshold are noise, not mathematics.
How do I evaluate a limit at infinity?+
Enter a very large x value and check that the output stabilises as you increase it further. A settling value indicates a horizontal asymptote.
What is an indeterminate form?+
An expression like 0/0 or ∞/∞ whose value cannot be read off directly. The limit may still exist; L'Hôpital's rule or algebraic simplification resolves it.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.