limit as x approaches 0 of 6(1/x-csc(x))

解答

x \to 0 lim (6 (\frac{1}{x} - csc (x)))

0

求解步骤

x \to 0 lim (6 (\frac{1}{x} - csc (x)))

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

= 6 \cdot x \to 0 lim (\frac{1}{x} - csc (x))

用 sin, cos 表示

= 6 \cdot x \to 0 lim (\frac{1}{x} - \frac{1}{sin ( x )})

化简

\frac{1}{x} - \frac{1}{sin ( x )} : \frac{sin ( x ) - x}{x sin ( x )}

= 6 \cdot x \to 0 lim (\frac{sin ( x ) - x}{x sin ( x )})

使用洛必达法则

= 6 \cdot x \to 0 lim (\frac{cos ( x ) - 1}{sin ( x ) + x cos ( x )})

使用洛必达法则

= 6 \cdot x \to 0 lim (\frac{- sin ( x )}{2 cos ( x ) - x sin ( x )})

代入值

x = 0

= 6 \cdot \frac{- sin ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

化简

6 \cdot \frac{- sin ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )} : 0

= 0

x \to 2 lim (\frac{2 - ∣ x ∣}{2 - x})

x \to 0 lim (\frac{g ( x ) - g ( 0 )}{x})

x \to 0 lim (\frac{x}{e ^{- \frac{7}{x}}})

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

x \to - 1 lim (2 x - 15)