This limit calculator estimates the value a function approaches as x heads toward a chosen point, including infinity. Limits are the idea on which all of calculus is built: they make sense of what happens to a function near a point where it might be undefined, and they define both the derivative, as the limit of a slope over a shrinking interval, and the integral, as the limit of a sum of shrinking pieces. The classic example is sine x over x, which is undefined at zero yet approaches 1 as x gets close, something a limit captures perfectly. You enter the function and the point that x approaches, and the calculator evaluates the function very close to that point from below and from above, reporting the left-hand value, the right-hand value, and the two-sided limit when they agree. When the two sides disagree, it tells you the two-sided limit does not exist, which is exactly what happens at a jump or a vertical asymptote. Enter inf or minus inf as the point to probe long-run behaviour and find horizontal asymptotes. The method is numerical, so it gives an accurate estimate rather than a symbolic proof, and the occasional awkward case near a singularity should be read with care. Enter a function and a point to see the limit straight away.
Use * for multiply and ^ for powers; pi and e are allowed, and inf or -inf for infinity. This is a numerical estimate. Estimate only.
For a finite point, the calculator evaluates the function at values extremely close to the point from each side and compares them. If both sides settle on the same number, that is the limit; if they differ, no two-sided limit exists. For infinity, it evaluates the function at very large values of x and reports the value it settles toward, or that it grows without bound.
For sine x over x as x approaches 0, the function is undefined at 0 but both the left and right values approach 1, so the limit is 1. For 1 over x as x approaches infinity, the value approaches 0.
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