→ Limit Calculator
Limits are the foundation of calculus: derivatives and integrals are both built on them. This calculator finds the limit of a function as x approaches a value by sampling points closer and closer from both sides — the same idea your textbook draws with arrows on a graph.
Type a function such as sin(x)/x or (x^2−1)/(x−1), set the point x approaches, and you will see the left-hand limit, the right-hand limit, whether the two-sided limit exists, and what the function itself does at the point.
How to use this calculator
Enter the function in terms of x, then the value x approaches. Both sides are sampled at x = a ± 0.1 down to ± 0.000001.
- Type your function f(x) using x, the functions sin, cos, tan, exp, ln, log, sqrt and abs, and the constants pi and e. Implied multiplication like 2x works.
- Enter the point a that x approaches — a plain number, or an expression like pi/2.
- Press Calculate. The calculator samples the function at a ± 0.1, ± 0.01 and so on down to ± 0.000001, from both sides.
- Read the verdict: if the left-hand and right-hand limits agree, that shared value is the limit. If they differ or explode, the limit does not exist — and the calculator says why.
Limit Worked Example
Find the limit of sin(x)/x as x → 0. At x = 0 the expression is 0/0 — undefined. But sampling nearby tells the story: at x = −0.1 the value is 0.998334, at −0.01 it is 0.999983; from the right, at +0.1 it is 0.998334 and at +0.01 it is 0.999983. Both sides close in on 1, so the limit is 1 even though f(0) itself does not exist. This is the famous sinc limit that makes the derivative of sin(x) work out so neatly.
Frequently asked questions
What is a limit in calculus?
A limit is the value a function approaches as x gets closer and closer to some point — even if the function is not actually defined at that point. The classic example is sin(x)/x as x approaches 0: at exactly 0 the expression is 0/0 and undefined, yet from both sides it closes in on 1, so the limit is 1.
Why does the calculator check the left and right sides separately?
A two-sided limit exists only when the left-hand limit and the right-hand limit are equal. For 1/x at 0 they are not: from the left the values plunge to −∞, from the right they soar to +∞, so the limit does not exist. Checking both sides is what catches jump discontinuities and vertical asymptotes.
Which functions and symbols can I use?
You can use + − * / and ^ for powers, brackets, and the functions sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, ln, log (base 10), sqrt, cbrt and abs. The constants pi and e work too, and multiplication can be implied — 2x and 3(x+1) are both fine. Angles are in radians.
Why does it say the limit could not be determined?
This calculator estimates limits numerically by sampling points ever closer to the target. Functions that oscillate infinitely fast near the point, like sin(1/x) at 0, have no stable value to converge to, and one-sided functions like sqrt(x) at 0 are undefined on one side — in those cases an honest 'could not determine' (or a one-sided answer by hand) is the correct outcome.
Is a numerical limit exact?
It is an estimate accurate to several decimal places for well-behaved functions, which is exactly what you need to check homework or build intuition. For a proof-grade answer you would still apply algebraic techniques — factoring, conjugates, L'Hôpital's rule or the squeeze theorem — but the numerical result tells you what those techniques should produce.