Limit Calculator

No. This is a numerical limit estimator. It does not parse expressions, simplify, or apply L'Hopital's rule symbolically. Instead, it evaluates one of 15 pre-defined function families (sin(x)/x, (1-cos x)/x, (1-cos x)/x^2, (e^x - 1)/x, ln(1+x)/x, x sin(1/x), sin(1/x), 1/x, |x|/x, x^2, e^x, ln(x), tan(x), a user-supplied polynomial, or a user-supplied rational function) at successively finer x values near the limit point and reads off the behavior. The trade-off: you cannot type an arbitrary formula, but every estimate is a real number you can verify by hand.

For a finite limit point a it samples f at x = a +/- 10^-k for k = 1 through 10, generating 10 one-sided values on each side. For limits at infinity it samples at x = +/- 10^k for the same range. It then looks for a stable plateau of three consecutive samples that agree within a small absolute and relative tolerance. The middle of that plateau is the converged value, rounded to four decimal places. Walking from coarse to fine and stopping at the first plateau avoids being fooled by catastrophic cancellation near the finest samples (a real problem for forms like (1 - cos x) / x^2, which look like 0 in floating point once |x| drops below about 1e-7).

Convergent: both one-sided sequences settle on the same finite value, so the two-sided limit exists. Infinite-positive or infinite-negative: both sides blow up in the same direction. DNE-asymmetric: the left and right limits exist separately but disagree (1/x at 0, |x|/x at 0, tan(x) near pi/2). DNE-oscillating: the sequence bounces without converging (sin(1/x) near 0). Indeterminate-form-0-0 or indeterminate-form-infinity-infinity: the direct substitution would give 0/0 or infinity/infinity AND the numerical evidence is inconclusive. Most classic indeterminate forms (sin(x)/x = 0/0) still resolve to a finite value and are reported as convergent.

Geometrically, for small positive x the area of the triangle with vertices (0, 0), (cos x, 0), (cos x, sin x) is less than the area of the circular sector of angle x in the unit circle, which is in turn less than the area of the triangle with vertices (0, 0), (1, 0), (1, tan x). Translating those inequalities gives cos(x) <= sin(x)/x <= 1/cos(x), and both ends go to 1 as x goes to 0, so sin(x)/x is squeezed onto 1. Numerically, the calculator confirms it: at x = 0.01 the value is already 0.99998333, at x = 0.0001 it is 1.00000000.

Approaching from the right (x = 0.1, 0.01, 0.001), 1/x climbs through 10, 100, 1000 toward +infinity. Approaching from the left (x = -0.1, -0.01, -0.001), 1/x falls through -10, -100, -1000 toward -infinity. Because the left and right one-sided limits disagree (one is +infinity, the other is -infinity), the two-sided limit does not exist and the calculator classifies it as DNE-asymmetric. Pick a one-sided direction (left or right) to see each one-sided limit on its own.