About the Limit Calculator
This limit calculator evaluates lim f(x) as x approaches a number, π/2, or positive or negative infinity. Choose a two-sided limit or a one-sided limit from the left (x → a⁻) or right (x → a⁺). Alongside the answer it reports both one-sided limits, the value of f at the point (or the indeterminate form such as 0/0 or ∞/∞ that direct substitution produces) and, for quotients, the derivatives used by L’Hôpital’s rule.
Use it to check calculus homework on classic limits like sin(x)/x, (1 − cos x)/x², (1 + 1/x)^x or √(x² + x) − x, to see whether a function has a removable discontinuity, a jump or a vertical asymptote, or to find horizontal asymptotes from limits at infinity. When the limit is a recognisable constant — a fraction, a multiple of π or e, a square root or a logarithm — it is shown in exact form next to the decimal.
The method is numerical: f is evaluated at points ever closer to a using 64-digit arithmetic, which avoids the rounding errors of ordinary calculators, and the trend is analysed to decide whether the values settle, blow up to ±∞ or oscillate. This is reliable for the functions met in calculus courses, but it is an estimate rather than a symbolic proof. Use x as the variable, ln for the natural log and radians for trig functions.
How to use the limit calculator
- 1Type the function using x, e.g. (x^2 - 9)/(x - 3).
- 2Enter the point x approaches — a number, pi/2, inf or -inf.
- 3Choose two-sided, left-hand or right-hand.
- 4Read the limit, the one-sided limits and the substitution check.
- 5For 0/0 or ∞/∞ forms, see the L’Hôpital derivatives in the method line.
Formula and method
A two-sided limit exists only when the left-hand limit (x → a⁻) and the right-hand limit (x → a⁺) exist and are equal. The calculator evaluates f at a − 10⁻ᵏ and a + 10⁻ᵏ for k = 2 to 17 (or at x = ±10ᵏ for limits at infinity) using 64-significant-digit arithmetic, then looks at the trend: values that settle give a finite limit (with a geometric tail extrapolation for slowly converging sequences), differences that stop shrinking signal divergence to ±∞, and values that never settle mean the limit does not exist.
If substituting x = a into a quotient gives 0/0 or ∞/∞, L’Hôpital’s rule says the limit equals the limit of the ratio of the derivatives, provided that limit exists. The calculator differentiates the numerator and denominator symbolically and shows that ratio. A limit can exist even where f(a) is undefined — that is the case of a removable discontinuity, such as (x² − 4)/(x − 2) at x = 2.
- a
- The point x approaches (a number or ±∞)
- x → a⁻ / a⁺
- Approach from the left / right
- L
- The limit value
- f′, g′
- Derivatives of numerator and denominator
Worked examples
The classic sin(x)/x as x → 0
Substituting 0 gives 0/0. L’Hôpital’s rule replaces the quotient with cos(x)/1, which equals 1 at x = 0, and the numerical values from both sides (0.99998…, 0.9999998…) agree.
Removable discontinuity: (x² − 4)/(x − 2) as x → 2
Factor the numerator: (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2, so the limit is 2 + 2 = 4 even though f(2) itself is undefined.
Limit at infinity: (1 + 1/x)^x
This famous limit defines Euler’s number. At x = 10⁶ the value is already 2.7182804…, and it approaches e = 2.718281828… as x grows.
One-sided limits at a vertical asymptote: 1/x at 0
From the right 1/x grows to +∞; from the left it falls to −∞. Because the one-sided limits differ, the two-sided limit does not exist.
(1 − cos x)/x² as x → 0
Using 1 − cos x ≈ x²/2 for small x, the ratio approaches 1/2. Ordinary floating-point calculators often get this wrong at very small x; 64-digit arithmetic avoids that.
Frequently asked questions
What does it mean for a limit to exist?+
The limit of f(x) as x approaches a exists if f(x) gets as close as you like to a single number L whenever x is close enough to a, from both sides. If the left and right limits differ, or the values blow up or oscillate, the limit does not exist.
When can I use L’Hôpital’s rule?+
Only when direct substitution gives an indeterminate form 0/0 or ∞/∞. Then lim f(x)/g(x) = lim f′(x)/g′(x), provided the new limit exists. Other forms such as 0·∞ or 1^∞ must first be rewritten as a quotient.
How do you find a limit at infinity?+
For rational functions compare the leading terms: equal degrees give the ratio of leading coefficients, a larger denominator degree gives 0. In general, divide by the highest power of x or use L’Hôpital’s rule on ∞/∞.
What is the difference between a one-sided and a two-sided limit?+
A one-sided limit only looks at x values on one side of a: from the left (x → a⁻) or from the right (x → a⁺). The two-sided limit exists only when both one-sided limits exist and are equal.
Can a limit exist if the function is undefined at that point?+
Yes. The limit describes values near a, not at a. (x² − 4)/(x − 2) is undefined at 2 but has limit 4 there, which is called a removable discontinuity.