Type f(x)
One box, total freedom — constants, decimals, fractions, parentheses, powers, π and e, and the functions sin, cos, tan, sqrt, ln, exp and abs. Any single letter works as the variable.
sin(3x)/x ✓ parses
01
The instrument
One input, zero buttons. The moment your limit parses, the screen normalizes it, the verdict lands in big type, and the work beneath is the complete written proof — substitution, 0/0 detection, factorization, cancellation, equivalences, degree rules, or the numeric table.
ONE VARIABLE · sin cos tan sqrt ln exp abs · ^ FOR POWERS · π AND e OK
x →
· OR inf / -inf
TRY
THE WORK — FULL PROOFLIVE
EXACT RATIONAL CORE·
0/0 DETECTION·
EQUIVALENCE TOOLBOX·
NUMERIC TABLE FALLBACK·
NOTHING LEAVES THE PAGE
02
How to use it
Six moves and you're fluent. The tool is designed to be read as much as used — every chip teaches, every step is copyable.
Set the approach
Where does x go? Any number — 3, −1, 0.5, pi/2, sqrt(2) — or inf and -inf for limits at infinity. The echo line renders the full lim notation live.
x → 3 · x → π/2 · x → ∞
Choose a side
Two-sided by default. Flip to x → c⁻ or x → c⁺ for one-sided limits — poles get exact sign analysis, and the verdict reports each side separately.
1/x at 0: left −∞ · right +∞
Watch it solve live
No equals button. The moment your input parses, the screen normalizes it and the work area begins writing — substitution first, then whatever the form demands.
keystroke → parsed → solved
Follow the written proof
Direct substitution, 0/0 detection, factorization at the root with every factor shown, cancellation, the equivalence swaps, degree rules at infinity — or the numeric table.
0/0 → factor → cancel → 6
Learn from the chips
Every example chip is a classic lesson: factoring, sin(3x)/x, 1 − cos x over x², the conjugate trick, degree rules, poles and the squeeze theorem.
tap a chip → watch the proof
case-file · worked exampleEXACT · NO ROUNDING
limit-calculator/
lim x→3 (x² − 9)/(x − 3)INPUT
0/0DIRECT · INDETERMINATE
x² − 9 = (x − 3)(x + 3)FACTOR
x + 3CANCEL
6SUBSTITUTE
6.001 / 5.999CHECK ✓
6LIMIT
1 LIMIT EVALUATED · 0 GUESSES MADE
03
Frequently asked questions
The questions every calculus student asks about limits — answered with worked numbers.
Q1How do I solve a limit step by step?
Try direct substitution first — if f(c) is defined, the limit is f(c). If you get 0/0, factor the numerator and denominator, cancel the common (x − c) factor, and substitute again. For trig and exponential forms use the standard equivalences; at infinity compare degrees; otherwise build a table of values. For (x² − 9)/(x − 3) at 3: substitution gives 0/0, factor to (x − 3)(x + 3)/(x − 3), cancel to x + 3, substitute to get 6. The calculator writes exactly these lines.
Q2What does lim x→c f(x) = L actually mean?
The ε–δ idea: f(x) can be forced as close to L as anyone demands, just by taking x close enough to c — from both sides, but never at c. The value f(c) is irrelevant to the limit and may not even exist. That is why (x² − 9)/(x − 3) has limit 6 at x = 3 even though the fraction is undefined exactly there.
Q3What is the 0/0 indeterminate form?
When substitution makes both the numerator and the denominator zero, the limit is genuinely hidden — 0/0 can equal anything: x/x → 1, x²/x → 0, sin(3x)/x → 3, (1 − cos x)/x² → 1/2. Indeterminate means the answer is decided by how fast each side approaches zero, not that no answer exists. Factor and cancel, or use the standard equivalences.
Q4How do I evaluate limits at infinity?
For rational functions use the degree rules: equal degrees → the ratio of leading coefficients, so (3x² + 1)/(5x² − 2) → 3/5; smaller degree on top → 0; bigger degree on top → ±∞. In general, divide everything by the highest power of x and let every 1/x term die. The tool displays this divide-by-xⁿ move line by line.
Q5When does a limit not exist?
Three classic ways: the two sides disagree — 1/x at 0 goes to −∞ from the left and +∞ from the right; unbounded growth — 1/x² at 0 blows up to +∞ (usually reported as an infinite limit rather than a failure); and oscillation — sin(1/x) at 0 never settles on any value. The solver announces which of the three verdicts your limit earns, with the evidence.
Q6What are the standard limit equivalences?
As u → 0: sin u ~ u, tan u ~ u, 1 − cos u ~ u²/2, e^u − 1 ~ u, and ln(1 + u) ~ u. Each means the ratio of the two sides tends to 1, so you may swap them in a quotient. Example: sin(3x)/x → 3x/x → 3. The solver applies exactly these swaps, and shows each one before continuing.
Q7How do one-sided limits work?
x → c⁻ approaches c from below, x → c⁺ from above. The two-sided limit exists only when both one-sided limits exist and agree. For √x at 0 the left-hand limit does not exist (the function is undefined for x < 0), so the two-sided limit fails while the right-hand limit is 0. Use the LEFT and RIGHT modes of the tool for these.
Q8What about L'Hôpital's rule?
For genuine 0/0 or ∞/∞ forms you may differentiate the numerator and denominator separately and retry. It is powerful, but the algebraic routes — factor and cancel, the equivalence toolbox, conjugates, degree rules — usually come first in class, and that is the order this solver follows. The numeric table serves as ground truth either way.
Q9Is it free, and does my input leave the page?
Completely free — no sign-up, no locked steps. All parsing and arithmetic run locally in your browser; nothing is recorded or transmitted, and the page keeps working offline once loaded.