Trigonometric limits

Trigonometric limits rest on one result: lim (x→0) (sin x)/x = 1, with x in radians. Check it on a calculator: sin 0.1/0.1 ≈ 0.998.

Why it is true: the sandwich theorem. If g(x) is always trapped between f(x) and h(x), and f and h both tend to the same number l, then g must tend to l too.

• In a circle of radius 1, comparing a small triangle, a sector and a bigger triangle gives sin x < x < tan x for small positive x.
• Dividing by sin x and flipping: cos x < (sin x)/x < 1.
• As x → 0, cos x → 1, so the trapped (sin x)/x → 1 as well.

Related results:

• lim (x→0) (tan x)/x = 1 and lim (x→0) x/(sin x) = 1.
• lim (x→0) (1 – cos x)/x = 0.
• Use 1 – cos x = 2 sin²(x/2) whenever 1 – cos appears.

The golden rule: the angle and the denominator must match. Make the bottom equal to the angle inside sin, and the angle must tend to 0.

• lim (x→0) (sin 5x)/x = lim 5 × (sin 5x)/(5x) = 5.
• lim (x→0) (tan 3x)/x = 3.
• lim (x→0) (sin 6x)/(sin 3x) = lim [(sin 6x)/(6x)] × [(3x)/(sin 3x)] × (6x/3x) = 1 × 1 × 2 = 2.

With 1 – cos.

• lim (x→0) (1 – cos x)/x² = lim 2 sin²(x/2)/x² = lim ½ × [(sin(x/2))/(x/2)]² = 1/2.
• lim (x→0) (1 – cos 2x)/x² = lim 2 sin²x/x² = 2.

When x does not tend to 0, shift it. lim (x→π) (sin x)/(π – x): put h = π – x, so h → 0. Since sin(π – h) = sin h, the limit is (sin h)/h → 1.

Split into parts. lim (x→0) (x + sin x)/x = lim [1 + (sin x)/x] = 1 + 1 = 2.