Investigate limits visually by zooming into function behavior near a point. Shows left/right limits, holes, and asymptotes.
Limit Explorer centers a function graph on an approach point and lets you zoom in to inspect what happens from smaller and larger x-values. Its separate left-hand and right-hand results help distinguish an ordinary limit, a jump, a hole, and unbounded behavior near a vertical asymptote. The important surprise is that the limit concerns nearby values, not necessarily the value at the point. A finite limit can exist even when the function is undefined there.
A concrete input and expected output from the current implementation.
Input
Function: (x^2 - 1)/(x - 1); point: x = 1
Expected output
Left-hand limit: 2; right-hand limit: 2; two-sided limit: 2; hole at (1, 2); f(1) is undefined.
For every x other than 1, factoring and canceling x - 1 reduces the expression to x + 1. Approaching 1 from either side therefore gives 2, while substitution into the original denominator gives zero.
After a function and approach point are selected, the graph can be examined at progressively smaller horizontal distances from that point. The left-hand result follows x-values below the target, while the right-hand result follows values above it. A two-sided limit exists only when those approaches agree; disagreement remains visible instead of being combined into one value.
The explorer treats nearby behavior separately from the function's value exactly at the target. A canceled factor can leave an undefined point even though surrounding values approach one finite height, producing a hole. If magnitudes increase without bound near the target, the display instead identifies asymptotic behavior. This distinction is essential when checking continuity.
Yes. The explorer examines values near the target, so both sides can approach the same number while the target itself is excluded. The example above approaches 2 even though its original expression divides by zero at x = 1. This is shown as a hole rather than as a failure of the limit.
No, not by itself. Matching sides establish the two-sided limit, but continuity at the point also requires the function to be defined there and to have that same value. A hole therefore fails continuity despite matching limits. The visual result can guide the check, while a proof may still require symbolic reasoning.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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