How to use the limit calculator
This calculator does not do symbolic algebra. It builds a rational function from the numerator and denominator you type, then evaluates it while stepping closer to a from the left and from the right: a numerical approximation. Arbitrary functions, including trigonometric, exponential and logarithmic ones and their compositions, are not supported.
If the denominator is not zero at a, the limit is simply the value obtained by substitution, so that exact value is shown, together with a reduced fraction when the coefficients are integers. If the numerator and denominator both vanish you have the 0/0 indeterminate form, so h shrinks from 1e-1 to 1e-6 while the two one-sided values are compared, and an estimate is reported. If they do not come close enough, the tool says convergence was not confirmed.
When only the denominator vanishes the value grows without bound. If both sides grow with the same sign the result is reported as +โ or โโ divergence; if the signs differ the limit does not exist. For x โ ยฑโ the leading degrees are compared: a higher numerator degree diverges, equal degrees give the ratio of the leading coefficients, and a higher denominator degree gives 0.
The table keeps every step of the approach so you can watch the values settle. Results print to 6 decimal places, and a coefficient or value counts as zero when its absolute value is below 1e-9, which avoids false readings from binary floating point. Degrees up to 12 are supported.
Frequently asked questions
No. This tool handles rational functions built from polynomials only, so trigonometric, exponential and logarithmic expressions cannot be entered.
Values are computed from both sides with h shrinking to 1e-6; when the two sides agree closely their average is reported as the estimated limit. It is a numerical estimate, not an algebraic result from factoring.
If the two sides do not settle on one value, the tool reports that the limit does not exist. If both grow with the same sign it reports +โ or โโ divergence instead.