Change, made visible

Limit calculator

Type a function and the point x approaches. The calculator gives both one-sided limits with error estimates and shows the values it used.

sin x/x as x → 0. Type any function and point; try 1/x, abs(x)/x or sin(1/x) at 0 to see limits that do not exist.

How the calculator finds a limit

A limit is decided by inputs near a, not at it[1]. The calculator evaluates f ever closer to a on each side and extrapolates the trend to distance zero, the same extrapolation it uses for derivatives[2]. It reports the limit from the left and from the right separately, each with an error estimate, and prints only the digits that estimate supports. The two-sided limit exists only when the two sides agree.

A worked example

The default is sin x/x at 0, which is 0/0 at the point itself. At x = 0.1 the function is 0.998334, at 0.01 it is 0.999983, at 0.001 it is 0.9999998: the values crowd toward 1 from both sides, and the calculator reports the limit 1 with an error estimate near 10−13. The table under “Show the evidence” lists those values. The proof that the limit is exactly 1 uses the squeeze theorem[1]; the calculator supplies evidence, not proof.

What it can and cannot tell you

  • A jump: abs(x)/x at 0 has left limit −1 and right limit 1, so there is no two-sided limit.
  • Unbounded growth: 1/x^2 at 0 is +∞ from both sides; 1/x at 0 goes up on one side and down on the other.
  • One side only: x^x and x*ln(x) at 0 are undefined for x < 0, so the calculator gives the right-hand limit (1 and 0) and says the left side is undefined.
  • No settling: sin(1/x) at 0 oscillates ever faster, and the calculator says the values do not settle.

Small results are printed as far as the error allows: x*sin(1/x) at 0 reads 0.00000 with an error bound below 10−6, not as a small nonzero number. A function could still wiggle between the samples in a way no finite set of samples detects; the proof that a limit is a particular number is the ε–δ argument, which you can play as a game on the page about the ε–δ definition.

Sources

  1. Gilbert Strang and Edwin “Jed” Herman (2016). Calculus Volume 1. OpenStax, Rice University. textbook Free, peer-reviewed. Used here: §2.2 The Limit of a Function, §2.5 The Precise Definition of a Limit, §3.1 Defining the Derivative, §3.2 The Derivative as a Function, §3.3 Differentiation Rules, §3.6 The Chain Rule, §4.1 Related Rates, §5.1 Approximating Areas, §5.2 The Definite Integral, §5.3 The Fundamental Theorem of Calculus.
  2. C. J. F. Ridders (1982). Accurate computation of F′(x) and F′(x)F″(x). Advances in Engineering Software 4(2), 75–76. doi:10.1016/S0141-1195(82)80057-0. primary