Limits of sine and cosine
For a tiny angle in radians, the arc, the straight chord and the height are almost equal. This is why sin x ≈ x for small x.
With x in radians: limx→0 (sin x)/x = 1, lim (1 − cos x)/x = 0, and lim cos x = 1, lim sin x = 0.
Formula: lim x→0 (sin x)/x = 1
Rule 1: Table of values — Small x in radians.
- Easy: x = 0.1: sin x/x = 0.99833.
- Medium: x = 0.01: sin x/x = 0.99998.
- Exam-type: x = 0.1: (1 − cos x)/x = 0.04996, tending to 0.
Rule 2: Use the limit — Adjust so that the angle in sin matches the denominator.
- Easy: lim(x→0) sin 3x/x = 3.
- Medium: lim(x→0) sin 5x/(2x) = 5/2.
- Exam-type: lim(x→0) (sin x)/(4x) = 1/4.
The limit is 1 only when x is in radians and the angle matches the denominator.
Using degrees. In degrees the limit is not 1.
Small angle: sine equals the angle.
Practice questions
- 1Find lim(x→0) sin 2x/x.
- 2Find lim(x→0) sin 7x/x.
- 3Find lim(x→0) sin 4x/(2x).
- 4Find lim(x→0) x/sin x.
- 5Find lim(x→0) sin 3x/sin 2x.
- 6Find lim(x→0) (1 − cos x)/x².
- 7Find lim(x→0) sin(x/2)/x.
- 8Find lim(x→0) (sin 6x)/(sin 3x).
Show answers (try them yourself first)
- 12
- 27
- 32
- 41
- 53/2
- 61/2
- 71/2
- 82
Tangent and other trigonometric limits
Once sin x/x → 1 is known, many other limits follow by rewriting.
Useful results: lim tan x/x = 1, lim tan kx/x = k, lim sin kx/x = k, lim sin ax/sin bx = a/b (x → 0). Use identities (like 1 − cos x = 2 sin²(x/2)) to match the pattern.
Formula: lim x→0 (tan x)/x = 1
Rule 1: Direct — Match the pattern.
- Easy: lim(x→0) tan 4x/x = 4.
- Medium: lim(x→0) tan 3x/tan 5x = 3/5.
- Exam-type: lim(x→0) sin 5x/tan 2x = 5/2.
Rule 2: Change of variable — Let t = x − a so that t → 0.
- Easy: lim(x→π) sin(π − x)/(π − x) = 1.
- Medium: lim(x→π/2) cos x/(π/2 − x) = 1.
- Exam-type: lim(x→π) sin x/(π − x) = 1.
If x → a (not 0), put t = x − a so that t → 0 first.
Applying the formula when the argument does not tend to 0.
Make the angle go to 0 first.
Practice questions
- 1Find lim(x→0) tan 2x/x.
- 2Find lim(x→0) tan 7x/tan 2x.
- 3Find lim(x→0) sin 3x/tan 2x.
- 4Find lim(x→π) sin x/(π − x).
- 5Find lim(x→0) (1 − cos 2x)/x².
- 6Find lim(x→0) (tan x − sin x)/x³.
- 7Find lim(x→π/2) cos x/(π/2 − x).
- 8Find lim(x→0) (sin 4x + tan 3x)/x.
Show answers (try them yourself first)
- 12
- 27/2
- 33/2
- 41
- 52
- 61/2
- 71
- 87