Rankwise

limx→0 sin x / x = 1

The limit of sin x / x as x tends to 0 is 1.

Proof

  1. For small x > 0, compare areas in the unit circle: sin x < x < tan x.
  2. Divide by sin x: 1 < x/sin x < 1/cos x, so cos x < sin x/x < 1.
  3. As x → 0, cos x → 1. Squeezed between two things that tend to 1, sin x/x → 1. It is even, so the left limit is the same.

Standard limits

LimitValue
limx→0 sin x / x1
limx→0 tan x / x1
limx→0 (1 − cos x) / x²1/2
limx→0 (1 − cos x) / x0
limx→0 (eˣ − 1) / x1
limx→0 (aˣ − 1) / xlog a
limx→0 log(1 + x) / x1
limx→0 (1 + x)^(1/x)e
limx→∞ (1 + 1/x)ˣe
limx→∞ (1 + a/x)ˣeᵃ
limx→a (xⁿ − aⁿ) / (x − a)n aⁿ⁻¹
limx→0 sin⁻¹x / x1
limx→0 tan⁻¹x / x1
limx→∞ sin x / x0
limx→0⁺ x log x0
limx→0⁺ xˣ1

Questions students ask

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

1.

How do you prove lim x→0 sin x / x = 1?

For small x > 0, compare areas in the unit circle: sin x < x < tan x. Divide by sin x: 1 < x/sin x < 1/cos x, so cos x < sin x/x < 1. As x → 0, cos x → 1. Squeezed between two things that tend to 1, sin x/x → 1. It is even, so the left limit is the same.

Use these next

Standard limitssin x/x, (eˣ − 1)/x, (1 + x)^(1/x) and every other standard limit, with proofs.Derivative formulasEvery standard derivative with its proof.Integration formulasEvery standard integral with a step-by-step proof.Maths formula sheetAlgebra to integrals, the formulas JEE Main uses most.