Limit Calculator
What a function approaches, evaluated from both sides.
Allowed: + - * / ^ ( ) and sin cos tan log ln sqrt abs exp pi e
A number, or inf / -inf
What a limit is
The limit of f(x) as x approaches a is the value f(x) settles on as x gets arbitrarily close to a. It says nothing about f(a) itself — the function need not even be defined there, which is exactly why limits are useful.
The classic case
sin(x)/x is undefined at x = 0, since both parts are zero. But evaluating either side of zero shows the values closing on 1, and that limit is what makes the derivative of sin(x) come out as cos(x).
One-sided limits
A two-sided limit exists only when both one-sided limits exist and agree. For 1/x at zero they do not: from the left the values fall without bound, from the right they rise without bound. The calculator reports the disagreement rather than inventing an answer.
This tool is numerical, not symbolic
It evaluates the function at points closer and closer to the target and reports the trend. That is strong evidence, not a proof, and it has the failure modes any numerical method has: floating-point cancellation very near the point, and functions that oscillate forever such as sin(1/x) at zero, where no finite table of values settles.
For a rigorous answer use the algebraic techniques — factor and cancel, rationalise, or L'Hôpital's rule — and treat this as the check on your work.
Limits at infinity
Enter inf or -inf. For a rational function the answer is decided by the degrees: equal degrees give the ratio of leading coefficients, a larger denominator degree gives 0, and a larger numerator degree diverges.