Limits Calculator
Solve limx→a f(x) Step-by-Step
Find the limit of any function as x approaches a number or infinity — from the left, right, or both sides — with a numerical approach table showing exactly how the value converges.
Enter your function
A Free Alternative to Mathway & Symbolab for Limits
Mathway and Symbolab hide their step-by-step limit solutions behind a paywall. This calculator is free end to end — the numerical approach table above and the worked examples below are visible with no subscription.
- ✓Numerical approach table free
- ✓Left/right/two-sided & infinity limits
- ✓Worked algebraic examples included
- ✓No signup, no paywall
- ✓Answer shown free
- !Step-by-step behind subscription
- ✓Answer shown free
- !Full steps behind subscription
This tool solves limits numerically rather than symbolically — see the FAQ below for what that means for your exam answers.
Standard Limit Formulas — Quick Reference
The results every Class 11-12, JEE and NEET student should have memorised, alongside what this calculator will show you numerically for each.
| As x → 0 or x → ∞ | Limit |
|---|---|
| sin(x) / x | 1 |
| (1 − cos x) / x | 0 |
| tan(x) / x | 1 |
| (eˣ − 1) / x | 1 |
| (aˣ − 1) / x | ln a |
| ln(1 + x) / x | 1 |
| (1 + 1/x)ˣ as x → ∞ | e |
| (xⁿ − aⁿ)/(x − a) as x → a | n·aⁿ⁻¹ |
| 1/x as x → ∞ | 0 |
Type any of these into the calculator above (e.g. sin(x)/x with a = 0) to see the numerical convergence for yourself.
Worked Examples — Full Algebraic Steps
The numerical table above shows convergence; here's the algebraic proof examiners expect for three common indeterminate-form limits.
limx→1 (x² − 1)/(x − 1)
- Direct substitution gives 0/0 — indeterminate, so factor first.
- x² − 1 = (x − 1)(x + 1)
- (x − 1)(x + 1) / (x − 1) = x + 1, for x ≠ 1
- limx→1 (x + 1) = 1 + 1 = 2
limx→0 sin(x)/x
- Direct substitution gives 0/0 — cannot factor, use the standard trig limit.
- This is one of the standard results: limx→0 sin(x)/x = 1
- Proved geometrically (squeeze theorem) in the Class 11 textbook — not by simplification.
- So the limit is 1, matching the numerical table above.
limx→∞ (3x² + 5x)/(x² − 1)
- Direct substitution gives ∞/∞ — indeterminate, divide every term by the highest power, x².
- (3 + 5/x) / (1 − 1/x²)
- As x → ∞, 5/x → 0 and 1/x² → 0
- Result: 3/1 = 3
How to Find a Limit
The limit of a function f(x) as x approaches a value a is the value f(x) gets closer and closer to — even if f(a) itself is undefined. This is why limits are the foundation of calculus: they let us analyze behavior at points like x = 0 in sin(x)/x, where direct substitution gives the indeterminate form 0/0.
limx→a f(x) = L
This calculator uses the numerical approach — the same method taught before formal limit rules: plug in values of x that get progressively closer to a (from both the left and right side), and see what value f(x) converges toward. If the left-hand limit and right-hand limit agree, the (two-sided) limit exists and equals that common value. If they disagree, the limit does not exist at that point.
For limits at infinity, the same idea applies — we substitute increasingly large values of x (or increasingly negative values for x → −∞) and observe what the function approaches. This is how you evaluate horizontal asymptote behavior, like 1/x → 0 as x → ∞.
Note: since this is a numerical approximation, indeterminate forms are evaluated by observing convergence rather than algebraic simplification (like factoring or L'Hôpital's Rule) — it's a great way to check your algebraic working, but always show the algebraic method in your exam answer.
← Back to Maths homework helpLimits Calculator — Questions Answered
At x = 0, sin(x)/x becomes 0/0, which is undefined — but the limit only cares about values *near* 0, not the value *at* 0. As x gets closer to 0 from either side, sin(x)/x gets closer and closer to 1, so the limit equals 1 even though the function itself is undefined at that exact point.
A two-sided limit does not exist when the left-hand limit and right-hand limit approach different values, when the function oscillates without settling on a value, or when the function grows without bound in a way that isn't consistently +∞ or −∞. This calculator shows the left and right values separately so you can see exactly where they diverge.
You can use standard arithmetic (+, −, ×, ÷, ^ for powers), and functions like sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, sqrt, ln (natural log), log (base 10), exp, abs, and pow. Use "x" as the variable, "pi" for π, and "e" for Euler's number.
Numerically it gets extremely close to the true value and is excellent for checking your work or building intuition, but it's an approximation, not an exact symbolic proof. For board exams, JEE and NEET, you should still show the algebraic method — factoring, rationalizing, or L'Hôpital's Rule — as this calculator won't produce that written proof for you.
Switch to the "x → ∞ or −∞" mode, enter your function, and choose the direction. The calculator substitutes increasingly large (or increasingly negative) values of x and shows how f(x) behaves — converging to a finite number, growing without bound (±∞), or approaching zero, which is exactly how horizontal asymptotes are identified.
Mathway and Symbolab show the final answer for free but put the step-by-step working behind a subscription. This calculator's numerical approach table and the worked algebraic examples on this page are free to view with no signup — though it solves numerically rather than symbolically, so for a formal exam proof you should still write out the algebraic steps shown in the examples above.
"Limit of a sum" usually refers to defining a definite integral as the limit of a Riemann sum — summing the areas of infinitely many infinitesimally thin rectangles as their number tends to infinity. That's a different (though related) topic from the point/infinity limits this calculator evaluates; it's covered under Integral Calculus rather than Limits & Continuity.