Evaluate left and right limits quickly. Visualize approach behavior with tables and charts. Understand function trends before exact substitutions every time.
A limit studies how a function behaves as x approaches a target value a. The calculator checks values near a from the left and right sides.
Left-hand limit: lim x→a⁻ f(x)
Right-hand limit: lim x→a⁺ f(x)
Two-sided limit exists when both one-sided limits approach the same number: lim x→a f(x) = L if lim x→a⁻ f(x) = lim x→a⁺ f(x) = L.
This page uses numerical approximation by evaluating f(a − h) and f(a + h) for smaller h values such as 1, 0.1, 0.01, and beyond.
| Example Function | Approach Point | Expected Limit | Reason |
|---|---|---|---|
| (x^2 - 1) / (x - 1) | 1 | 2 | Common removable discontinuity after factor cancellation. |
| sin(x) / x | 0 | 1 | Classic trigonometric limit near zero. |
| 1 / x | 0 | DNE | Left and right sides diverge differently. |
| abs(x) / x | 0 | DNE | Left side is -1 and right side is 1. |
A function limit describes the value f(x) approaches as x gets very close to a chosen point. The actual function value at that point may match, differ, or even be undefined.
Limits focus on nearby behavior, not only the exact point. A hole or removable discontinuity can leave the function undefined while surrounding values still approach one clear number.
The left-hand limit checks values approaching from smaller x values. The right-hand limit checks larger x values. A two-sided limit exists only when both sides approach the same result.
DNE means the limit does not exist numerically. This happens when one-sided limits disagree, values oscillate, or the function grows in different infinite directions near the target point.
Yes. This calculator supports common functions like sin, cos, tan, sqrt, exp, log, ln, and abs. Use x as the variable and standard mathematical notation.
Numerical approximation helps you inspect behavior near a target when symbolic simplification is difficult. Tables and graphs often reveal convergence patterns, discontinuities, or opposing one-sided trends quickly.
The h value is the small distance from the approach point. The calculator evaluates f(a − h) and f(a + h) for shrinking h values to estimate the limit.
No. The graph intentionally skips the exact approach point during plotting when needed. This makes holes, jumps, and undefined behavior easier to inspect visually around the target.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.