Limit Calculator

This calculator evaluates a one-sided-then-combined limit for one of three built-in function families — a rational function ax over (bx + 1), a trigonometric ratio a·sin(x) over bx, or an exponential ratio a·(e^(bx) − 1) over x — by numerically probing the function at a point just below and just above the value x approaches, rather than manipulating the algebra symbolically. The left and right probes sit only one hundred-millionth away from the approach point on either side, close enough that if both probes land within a thousandth of each other, the calculator reports the limit as existing and averages the two. At the calculator's default settings, with x approaching 0 under the rational function, neither coefficient has any effect on the reported limit at all: the numerator of ax over (bx + 1) is proportional to x itself, so as x shrinks toward zero the numerator vanishes regardless of what a equals, while the denominator settles at exactly 1 regardless of what b equals.

The coefficients only start to matter once x approaches some value other than zero, or once you switch to the trigonometric or exponential function type, where a zero-over-zero form is resolved with a hardcoded algebraic shortcut instead of the tiny-probe method. This page does not attempt to: evaluate a limit for any function you type in yourself — only these three pre-built families — and it has no notion of a limit at positive or negative infinity, only at whatever finite number you enter for the value x approaches.