Limit Calculator

This calculator numerically estimates the limit of any function as x approaches a value you choose, evaluating from both sides so you can see the left limit, the right limit and whether a two-sided limit exists. The concept of a limit is the foundation of calculus. A limit asks: as x gets closer and closer to some value c, what value does f(x) approach? The function does not need to be defined at c itself; it is the behaviour in the neighbourhood of c that matters. Limits are used to define the derivative (the slope of a curve at a point) and the definite integral (the total area under a curve), and they appear wherever a function has a removable discontinuity, a jump, or an asymptote. This calculator evaluates f(x) at points increasingly close to c from the left and from the right, checking whether those values converge to a common number. If the left limit and right limit agree to four decimal places, the two-sided limit is reported; if they diverge or the function grows without bound, that is flagged. You type the function (for example (x*x-1)/(x-1) or sin(x)/x), enter the value c that x approaches, and all three limit estimates appear immediately. Supported functions include sin, cos, tan, log, exp, sqrt and any composable expression using the JavaScript Math object. Approach values that land exactly on an asymptote (such as x approaching 0 in 1/x) will show divergence correctly. The tool is suited to students learning calculus at NCEA Level 3 or university level. Numerical limits are approximations and should be confirmed algebraically where an exact answer is required.

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Evaluates at c ±0.001 and c ±0.0001 to detect convergence. Not suitable for wildly oscillating functions near c (e.g. sin(1/x) as x approaches 0).