Educational tool; results are estimates for learning, verify with your textbook or instructor.
This tool probes $\lim_{x\to a}f(x)$ or the behavior of $f$ as $x\to\pm\infty$ by evaluating a constrained expression at a ladder of nearby points. It reports a guessed limit and the last samples from each side (or from large $|x|$).
The guess is meaningful when the samples from both sides settle near the same number, or when the large $|x|$ values settle. Agreement among samples is evidence, not a proof. A formal limit still needs the epsilon delta guarantee, or an algebra argument that removes a hole.
This is not a computer algebra system and not an epsilon delta writeup. Oscillation, a jump, or a vertical blow up can fool any finite ladder. The default $(x^2-1)/(x-1)$ as $x\to 1$ is a hole you can fill: samples should approach $2$.
Allowed pieces match the derivative tool: numbers, $x$, plus, minus, times, divide, powers, parentheses, and sin cos tan sqrt abs log exp pi e.
How to use the probe
Type $f(x)$. Choose a finite approach $x\to a$ or an infinite approach. For a finite approach, enter $a$. Press Estimate limit. The page samples a fixed ladder of distances, from $10^{-1}$ down to $10^{-6}$ on each side, or a ladder of large $|x|$ values.
Keep the default run until you see a number near $2$ with a small gap between left and right. Then change the formula to something with a jump, such as $\mathrm{abs}(x)/x$ as $x\to 0$, and watch the gap stay large.
How to read the output
The large number is an average of the nearest left and right samples (or the last large $|x|$ sample). The note prints those neighbors and the gap. A tiny gap plus stable samples is the happy case.
A large gap means the two sides do not agree. In analysis that usually means the two sided limit does not exist, even if each one sided limit exists. The page is showing you that disagreement as numbers.
Samples that become non finite drop out. If too many drop out, the tool refuses the estimate and asks you to check the domain.
Common pitfalls
A removable hole can still produce a huge intermediate value if you land exactly on the hole. The sampler avoids $x=a$ on purpose. If you rewrite the formula badly, you can still hit a singularity on the ladder.
Rapid oscillation such as $\sin(1/x)$ near $0$ can look settled at an unlucky sample and wild at the next. Finite samples do not replace a picture of the oscillation.
As $|x|$ grows, a polynomial dominates, but a ratio of huge terms can overflow before it settles. Overflow is not the same as “the limit is infinity” in a careful course, though it is a hint to look at degrees.
Related reading
The definition is unpacked in epsilon delta intuition and at limit, epsilon delta definition, and continuity. Once you have a candidate for $L$, the epsilon delta finder asks a sharper question. You also type epsilon, and the page searches for a large delta that keeps the sampled outputs inside that tolerance. That search is still a numerical check, not a proof. After a limit settles, slopes are in what a derivative actually measures and on the numerical derivative calculator.
The map is calculus foundations. A path is the AP Calculus AB bridge. Accumulation after limits: Riemann sum calculator.
