limit as x approaches pi/3+of (1-cos(3x))/(sin(3x))

解

x \to \frac{π}{3} + lim (\frac{1 - cos ( 3 x )}{sin ( 3 x )})

- \infty

解答ステップ

x \to \frac{π}{3} + lim (\frac{1 - cos ( 3 x )}{sin ( 3 x )})

基本的な三角関数の公式を使用する:

\frac{1}{sin ( x )} = csc (x)

= x \to \frac{π}{3} + lim (csc (3 x) (1 - cos (3 x)))

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

不定形式の場合を除く

= x \to \frac{π}{3} + lim (csc (3 x)) \cdot x \to \frac{π}{3} + lim (1 - cos (3 x))

x \to \frac{π}{3} + lim (csc (3 x)) = - \infty

x \to \frac{π}{3} + lim (1 - cos (3 x)) = 2

= (- \infty) \cdot 2

無限大の特性を適用する:

c \cdot (- \infty) = - \infty

= - \infty