limCalculus · Limits
Limit Calculator that names the form first
Type a limit and we'll tell you what kind it is — 0/0, ∞/∞ or neither — then solve it with the method that fits, from both sides.
- Free steps
- Graph
- Checked answers
Try it
Zoom into the point. Do both sides agree?
A limit asks what f(x) approaches as x gets closer to a point — from both sides — never what happens exactly at it.
| x | f(−x) | f(x) |
|---|---|---|
| 1 | 0.841471 | 0.841471 |
| 0.1 | 0.998334 | 0.998334 |
| 0.01 | 0.999983 | 0.999983 |
| 0.001 | 1 | 1 |
Undefined at x = 0, yet both sides close in on 1 — that common value is the limit.
Why this limit calculator is different
01
Names the form first
Direct substitution comes first. If it gives 0/0, (∞)/(∞), 0×∞, ∞-∞ or 1^(∞), you see the form and every method that applies, with one line on why.
02
Two methods, one answer
Limits settled by L'Hôpital or a standard limit also get the Taylor-series route, so you see why (1 - cos x)/(x^2) → 1/2, not just that it does.
03
An honest check
A table of values from both sides checks the algebra independently — with 64-digit arithmetic when ordinary decimals cancel or overflow. Oscillating limits are confirmed numerically; if a check is impossible, the steps say why.
What is a limit calculator?
A limit calculator finds lim _x→ a f(x) — the value f(x) approaches as x gets close to a — and shows how. The limit **never uses the value at a itself**, which is why ( sin x)/(x) has the limit 1 at x = 0, where it is not even defined.
Often you can just substitute. When substitution gives an indeterminate form like 0/0 or (∞)/(∞), the expression must be rewritten first: factor and cancel, multiply by a conjugate, use a standard limit or a Taylor series, divide by the dominant power, or apply L'Hôpital's rule. A two-sided limit exists only if the left-hand and right-hand limits agree.
Limits are the foundation of calculus — derivatives and integrals are both defined by them — and appear in CBSE Class 11 "Limits and Derivatives", A-level differentiation from first principles, AP Calculus AB/BC, IB Mathematics and JEE. This limit calculator shows the method a teacher would choose, step by step.
How to use the limit calculator
- 1Type the function, e.g. `(x^2 - 4)/(x - 2)`, and enter the value x approaches: a number, `pi`, or `inf` / `-inf`. Or type the whole limit: `lim x->2 (x^2-4)/(x-2)`, or paste LaTeX like `\lim_{x \to 0} \frac{\sin 3x}{x}`.
- 2Choose Both sides, or Left (−) / Right (+) for a one-sided limit such as lim _x→0^+ x ln x. Cube roots are real for negative x, so limits at -8 involving root 3 of (x) work from both sides.
- 3Read the first step: what substitution gives and, if the form is indeterminate, which methods apply. Then follow the chosen method line by line.
- 4Open Table to watch f(x) approach the limit from each side, and Graph to see the hole (hollow point) or the asymptote.
Methods for evaluating limits
The calculator tries these in the order a teacher would:
Direct substitution
Works whenever f is continuous at a — polynomials everywhere, quotients where the denominator is not zero.
Factor and cancel
For 0/0 with polynomials: both parts share the factor (x - a), which cancels because x ≠ a in a limit.
Multiply by the conjugate
Clears a square root that causes 0/0 or ∞ - ∞, using (A - B)(A + B) = A^2 - B^2.
Standard limits and Taylor series
Replace each function by its series near the point; the lowest surviving power on top and bottom decides the limit. This settles ( cos x - 1 + x^2/2)/(x^4) = 1/24, which needs four rounds of L'Hôpital.
Dominant terms at infinity
Divide top and bottom by the highest power of x; every smaller term tends to 0. Exponentials beat powers, which beat logarithms.
L'Hôpital's rule
Differentiate the top and the bottom separately (not the quotient rule) and try again; 0×∞, 1^∞ and 0^0 forms are rewritten first.
Limit calculator worked example: (√(x+4) − 2)/x
- 01
Substitute x = 0: the top is √(4) - 2 = 0 and the bottom is 0 — the indeterminate form 0/0.
(√(0+4)-2)/(0) → 0/0 - 02
A square root causes the problem, so multiply top and bottom by the conjugate √(x+4) + 2.
(√(x+4)-2)/(x)×(√(x+4)+2)/(√(x+4)+2) = (x)/(x(√(x+4)+2)) - 03
Cancel the common factor x (allowed, since x ≠ 0 while approaching), then substitute.
lim _x→0(1)/(√(x+4)+2) = 1/4 - 04
The table agrees: f(-0.001) ≈ 0.250016 and f(0.001) ≈ 0.249984, closing in on 0.25 from both sides.
Mistakes the limit calculator catches
Saying 0/0 means "no limit"
0/0 only means substitution failed. The limit can be any number — lim _x→0( sin x)/(x) = 1, lim _x→0(x^2)/(x) = 0 — so rewrite the expression and try again.
Using the quotient rule in L'Hôpital's rule
L'Hôpital differentiates the numerator and the denominator separately: (f')/(g'), not (f/g)'. And it only applies to 0/0 or (∞)/(∞).
Ignoring the two sides
lim _x→01/x does not exist: from the right the values go to +∞, from the left to -∞. Check both sides whenever a denominator changes sign.
Treating 1^(∞) as 1
(1 + 1/x)^x → e, not 1. A base tending to 1 raised to a growing power is indeterminate — use e^( lim g(f-1)) or take logarithms.
Limit calculator FAQ
Is this limit calculator free?+
Yes. Every step, the table of values, the graph and the practice problems are free for typed limits, with no sign-up and no adverts. Photo questions are read by AI and have a daily allowance, but the steps for typed problems are never locked behind a paywall.
Can it do one-sided limits?+
Yes. Choose Left (−) or Right (+), or type `0+` / `0-` as the point. For a two-sided limit the calculator works out both sides whenever they could differ — a denominator changing sign, an absolute value — and reports "does not exist" when they disagree.
Does it use L’Hôpital’s rule?+
Yes, for 0/0 and (∞)/(∞) forms, showing each derivative. It prefers simpler methods when they work — factor and cancel, conjugates, standard limits, dominant terms — and adds a Taylor-series check, because those are what exams usually expect. When L'Hôpital would need many rounds, Taylor series is used instead.
How is the answer checked?+
Independently of the algebra: the function is evaluated at points ever closer to the target from each side, and those values must settle on the answer before the Verified stamp appears. When ordinary decimals lose accuracy or overflow, the check is repeated with 64-digit arithmetic.
Why is there sometimes no Verified stamp?+
Because the check must be honest. If the values approach too slowly to settle at six decimal places, or are too extreme to sample, the steps end with a short note explaining why the numerical check could not confirm the answer. Oscillating limits, such as sin 1/x at 0, are confirmed by sampling.
What are indeterminate forms?+
Results of substitution that do not decide the limit: 0/0, (∞)/(∞), 0×∞, ∞ - ∞, 1^(∞), 0^0 and ∞^0. Each one needs the expression to be rewritten — factorised, rationalised, expanded in a series or turned into a quotient — before the limit can be found.
How do I enter infinity or π?+
Type `inf` or `infinity` (or `-inf`) in x approaches, and `pi` for π, for example `pi/2`. You can also type the whole limit in one line, such as `lim x->oo (1+1/x)^x`, which gives e by the 1^(∞) rule, or `lim x->0+ x ln(x)` for a one-sided limit.
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