Most important formula: lim_{x→0} (sin x)/x = 1（x is in radians）.
Derived Formulas
- lim_{x→0} x/(sin x) = 1
- lim_{x→0} (tan x)/x = 1
- lim_{x→0} (1 − cos x)/x² = 1/2
- lim_{x→0} (1 − cos x)/x = 0
Example ①
(1) lim_{x→0} (sin 3x)/x (2) lim_{x→0} (sin 2x)/(sin 5x)
Solution:(1) 3·(sin 3x)/(3x) → 3 (2) 2/5
(1) lim_{x→0} (sin 3x)/x (2) lim_{x→0} (sin 2x)/(sin 5x)
Solution:(1) 3·(sin 3x)/(3x) → 3 (2) 2/5
(1 − cos x)/x²
Use 1 − cos x = 2 sin²(x/2).
(1 − cos x)/x² = (1/2){sin(x/2)/(x/2)}² → 1/2
Example ②
lim_{x→0} (1 − cos 2x)/x²
Solution:u=2x： (1−cos u)/(u/2)² = 4·(1−cos u)/u² → 4·(1/2) = 2
lim_{x→0} (1 − cos 2x)/x²
Solution:u=2x： (1−cos u)/(u/2)² = 4·(1−cos u)/u² → 4·(1/2) = 2
Key Points
- (sin x)/x → 1（radians!）
- (sin ax)/x → a、(tan ax)/x → a
- (1−cos x)/x² → 1/2
- Factor out constant multiples to match the formula
Practice Problems
- lim_{x→0} (sin 5x)/(3x)
- lim_{x→0} (tan x)/(sin 2x)
- lim_{x→0} (1 − cos x)/(x sin x)
Answers and Explanations
- Answer:5/3
(5/3)·(sin 5x)/(5x) → 5/3。 - Answer:1/2
(tan x / x)/(sin 2x / x) → 1/2。 - Answer:1/2
{(1−cos x)/x²}·{x/sin x} → 1/2·1 = 1/2。