Enter a function and the value x approaches. The calculator finds the exact limit with the rule used at each step (substitution, factoring, conjugates, L'Hôpital's rule, dominant terms), checks both sides, and shows a table of values and a graph so you can see the limit happen.
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Limitexact
Step by step
Numeric approach
Values of f closer and closer to the point. They should settle toward the limit.
Graph near the point
Dashed lines mark the point and the limit value; a hollow dot marks the limit.
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How to use the Limit Calculator
Type the function, for example sin(x)/x or (1 + 1/x)^x. Use ^ for powers and sqrt() for roots.
Enter the point x approaches: a number, an expression like pi/2, or press ∞ / −∞.
Choose the direction: two-sided, from the left or from the right.
Press Calculate limit to see the answer, the steps, the table of values and the graph.
Copy the result if you need it elsewhere.
Strategy for evaluating a limit
Substitute directly. If f is continuous at a (polynomials, roots, exponentials, sin and cos on their domains), the limit is just f(a): lim x→2 of x² + 3x = 10.
If you get a nonzero number over 0, the function blows up. Check the sign on each side: 1/x goes to −∞ from the left and +∞ from the right, so the two-sided limit does not exist.
If you get 0/0, simplify: factor and cancel ((x² − 1)/(x − 1) → x + 1 → 2), multiply by a conjugate for square roots, or use L'Hôpital's rule.
At ±∞, keep only the dominant terms. Exponentials beat powers, and powers beat logarithms: x³/eˣ → 0 and ln(x)/x → 0.
For variable powers like (1 + 1/x)ˣ, write f = e^(g·ln h) and find the limit of the exponent: x·ln(1 + 1/x) → 1, so the limit is e.
Standard limits worth memorising
Limit
Value
Comment
lim x→0 sin(x)/x
1
Basis of the derivative of sin; x in radians
lim x→0 (1 − cos x)/x²
1/2
From cos x ≈ 1 − x²/2
lim x→0 (eˣ − 1)/x
1
Derivative of eˣ at 0
lim x→0 ln(1 + x)/x
1
Derivative of ln at 1
lim x→∞ (1 + 1/x)ˣ
e ≈ 2.71828
Definition of e; (1 + k/x)ˣ → eᵏ
lim x→0⁺ x·ln x
0
Powers beat logarithms
lim x→0⁺ xˣ
1
0⁰ form, via e^(x ln x)
lim x→∞ xⁿ/eˣ
0
For every n; exponentials dominate
lim x→∞ (√(x² + x) − x)
1/2
∞ − ∞, solved with the conjugate
lim x→0 x·sin(1/x)
0
Squeeze theorem: |x sin(1/x)| ≤ |x|
One-sided limits
The left-hand limit lim x→a⁻ uses only values x < a and the right-hand limit lim x→a⁺ uses only x > a. The two-sided limit exists exactly when both one-sided limits exist and are equal. One-sided limits are needed at the edge of a domain (ln x and √x only make sense for x → 0⁺), at jumps, and at vertical asymptotes such as tan x at π/2.
Frequently asked questions
What does a limit mean?
The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to when x gets close to a, without needing x to equal a. For example sin(x)/x is undefined at x = 0, but its values approach 1, so the limit is 1.
When does a limit not exist?
A two-sided limit does not exist when the left-hand and right-hand limits are different (|x|/x at 0 gives −1 and 1), when the function oscillates without settling (sin(1/x) at 0 or sin(x) at infinity), or when the function grows without bound with opposite signs on the two sides (1/x at 0). A limit of +∞ or −∞ is also technically non-existent but describes how the function behaves.
What is L'Hôpital's rule?
If f(x)/g(x) gives the indeterminate form 0/0 or ∞/∞ at the point, then lim f(x)/g(x) = lim f′(x)/g′(x), provided the right-hand limit exists. The calculator differentiates the numerator and denominator separately and repeats if the form is still indeterminate.
How do you find limits at infinity of rational functions?
Compare the degrees of the numerator and the denominator. If the numerator has the lower degree, the limit is 0. If the degrees are equal, the limit is the ratio of the leading coefficients, so (3x² + 2)/(5x² − x) → 3/5. If the numerator has the higher degree, the function goes to +∞ or −∞.
What are indeterminate forms?
Expressions like 0/0, ∞/∞, 0·∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰ do not have a fixed value; the limit depends on how fast each part changes. The calculator rewrites them (as a quotient, a single fraction, with a conjugate, or as e raised to a logarithm) until a rule gives the value.
Why does the answer say numeric estimate?
If no symbolic rule finishes the problem, the limit is estimated from function values very close to the point and, where possible, matched to an exact constant such as 1/3, π or ln 2. Treat it as a strong hint rather than a proof and check the table and the graph.
Are angles in radians?
Yes. Trigonometric functions use radians, which is what calculus limits like sin(x)/x → 1 assume. Enter π as pi, for example tan(x) as x → pi/2.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.