Numerical Limit Estimate Calculator
Estimate the limit of a rational function as x approaches a point by evaluating it from both sides — great for removable 0/0 forms.
How it works
lim(x→a) f(x) estimated from f(a − h) and f(a + h) with h = 1e-5 · a two-sided limit exists only if both sides agree
Many interesting limits produce the indeterminate form 0/0 when you plug the point in directly. Evaluating the function a hair to the left and right instead reveals the value the function is heading toward, sidestepping the undefined point. If the two one-sided estimates disagree, the two-sided limit does not exist — a jump or a pole is present.
Worked example
For (x² − 1)/(x − 1) as x → 1, direct substitution gives 0/0. Evaluating at 0.99999 and 1.00001 gives about 1.99999 and 2.00001, so the limit is 2 — matching the algebra, since the expression simplifies to x + 1.
Sources: Limit of a function — Wikipedia
Frequently asked questions
Why can't I just plug in the point?
For a removable discontinuity the function is literally undefined at the point (0/0), so substitution fails. The limit asks what value the function approaches nearby, which is exactly what sampling both sides reveals.
What does "does not exist" mean here?
If the left-hand and right-hand estimates differ by more than a small tolerance, the function approaches different values from each side — a jump discontinuity or an asymptote — so no single two-sided limit exists.
Is a numerical estimate exact?
It is an approximation, accurate to a few decimals for well-behaved functions. Near a genuine pole or a highly oscillatory point the numbers can mislead, so confirm surprising results algebraically.
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This calculator is for educational purposes. Double-check important results before acting on them.