Limit calculator: one-sided limits and indeterminate forms

What a limit says, and what the indeterminate forms are for

The limit of f(x) as x tends to a is the value f(x) gets close to as x gets close to a, whatever happens at a itself. The function need not even be defined there: sin x / x cannot be computed at 0, and yet its values close in on 1 from both sides. That is the point of a limit — it describes the approach, not the arrival.

Approaching from the left and from the right are two separate questions. When both give the same value, that is the limit; when they differ, the limit does not exist. 1/x near 0 runs to −∞ from the left and to +∞ from the right, which is exactly the signature of a vertical asymptote. At ±∞ there is only one side to come from.

An indeterminate form is not an answer but a warning that substitution tells nothing. 0/0, ∞/∞, ∞ − ∞, 0 · ∞, 1^∞, 0⁰ and ∞⁰ can each end in any value, and the work is to rewrite the expression until the form goes away: factorising, rationalising, a notable limit or de l'Hôpital's rule. The form shown here says which kind of rearrangement is needed.

Common mistakes

Reading 0/0 as 0 or as 1. It is a form, not a number: (x² − 1)/(x − 1) tends to 2 as x → 1, sin x / x to 1, and x²/x to 0.
Treating 1^∞ as 1. (1 + 1/x)^x tends to e, not to 1, because the base gets close to 1 exactly as fast as the exponent grows.
Giving a two-sided limit where the one-sided limits differ: lim x→0 of 1/x does not exist, because the two sides go to −∞ and +∞.
Concluding that a limit does not exist because the function is not defined at the point: the limit only looks at the values near it.

Frequently asked questions

When does a limit not exist?

When the left-hand and right-hand limits differ, as for 1/x at 0, or when the function keeps oscillating without settling, as sin(1/x) near 0 or sin x at infinity.

What are the indeterminate forms?

0/0, ∞/∞, ∞ − ∞, 0 · ∞, 1^∞, 0⁰ and ∞⁰. Substituting the point gives one of them, which says nothing about the limit: the expression has to be rewritten until the form disappears.

What is the limit of sin x / x as x tends to 0?

1. It is the first of the notable limits, and it is a 0/0 form: the sine of a small angle is almost equal to the angle itself, so the ratio closes in on 1 from both sides.

How does the calculator work out a limit?

It evaluates the function at points closer and closer to the one asked, and reads the limit where the values stop changing, before rounding error takes over. A value that settles is written in exact form when it is recognised — 1/2, e, √2 — and one that keeps growing is reported as ±∞.

How this calculation works

lim x→a f(x) = L when f(x) gets arbitrarily close to L as x gets close to a, a excluded. Left-hand limit: x → a⁻, with x < a; right-hand: x → a⁺, with x > a. The limit exists when both one-sided limits exist and agree. Notable limits: lim x→0 sin x / x = 1, lim x→0 (1 − cos x)/x² = 1/2, lim x→0 (eˣ − 1)/x = 1, lim x→0 ln(1 + x)/x = 1, lim x→±∞ (1 + 1/x)^x = e. Numerically, f is evaluated at a ± h with h = 10^(−k/2), k = 2, 3, …, 20, or at x = ±10^(k/2) towards infinity; the limit is read where the differences between consecutive values stop shrinking.

Study of a function

Domain, sign, limits, asymptotes, turning points, concavity and graph, in one go.

Domain of a function

The conditions of existence and the domain written as intervals.

Asymptotes

Vertical, horizontal and oblique asymptotes, with the limits that give them.

Derivatives

The derivative of a function and the rules used to get it.