Limit Calculator
Numerically estimate the limit of a function as x approaches a chosen value by sampling the function from both sides.
Function f(x)
Left and right estimates agree
How does this calculator work?
A limit describes the value a function approaches as x nears c. This calculator samples f(x) at points just left and right of c (c ± 10⁻ᵏ). If both sides converge to the same number, that average is the two-sided limit; if they differ, the limit does not exist.
Formula
How this is calculated
Pick a function f(x) and the value c that x approaches. The calculator evaluates f at points c − 10⁻ᵏ and c + 10⁻ᵏ for k = 1 through 6, marching steadily closer to c from the left and from the right. The value reached on each side at the smallest finite step is reported as the one-sided (left and right) limit estimate.
If the left and right estimates agree within a small tolerance (about 1e-4 scaled by the magnitude), their average is reported as the two-sided limit. If they disagree — as with |x|/x at 0, where the left side gives −1 and the right side gives +1 — the two-sided limit does not exist. This mirrors the formal definition: a two-sided limit exists only when both one-sided limits exist and are equal.
This is a numerical estimate, not a symbolic proof. Functions that oscillate infinitely (like sin(1/x) near 0) or diverge to infinity can fool the sampling, and floating-point rounding limits precision near very steep regions. For removable discontinuities such as sin(x)/x at 0 or (x²−1)/(x−1) at 1, the method recovers the true limit (1 and 2 respectively) cleanly.
Frequently asked questions
A two-sided limit exists only if the left and right one-sided limits are both finite and equal. When they differ (a jump) or one side diverges, the calculator reports that the limit does not exist.
No. It is a numerical estimate found by sampling the function very close to c. It works well for continuous functions and removable discontinuities, but can be misled by wildly oscillating or rapidly diverging functions.
This tool evaluates limits as x approaches a finite value c. To approximate behaviour at infinity, enter a large c and inspect the trend, though dedicated end-behaviour analysis is more reliable.
Also known as
TG we-Calculate Editorial Team. (2026). Limit Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/limit-calculator
TG we-Calculate Editorial Team. "Limit Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/limit-calculator.
TG we-Calculate Editorial Team, "Limit Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/limit-calculator
@misc{wecalculate_limit_calculator, title = {Limit Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/limit-calculator}}, year = {2026}, note = {TG we-Calculate} }
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