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limit as x approaches 0 of 7csc(x)-7/x

解答

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

解答

0

求解步骤

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

7 csc (x) - \frac{7}{x} = 7 csc (x) (1 - \frac{\frac{7}{x}}{7 csc ( x )})

= x \to 0 lim (7 csc (x) (1 - \frac{\frac{7}{x}}{7 csc ( x )}))

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

= 7 \cdot x \to 0 lim (csc (x) (1 - \frac{\frac{7}{x}}{7 csc ( x )}))

化简

csc (x) (1 - \frac{\frac{7}{x}}{7 csc ( x )}) : csc (x) - \frac{1}{x}

= 7 \cdot x \to 0 lim (csc (x) - \frac{1}{x})

用 sin, cos 表示

= 7 \cdot x \to 0 lim (\frac{1}{sin ( x )} - \frac{1}{x})

化简

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

= 7 \cdot x \to 0 lim (\frac{x - sin ( x )}{x sin ( x )})

使用洛必达法则

= 7 \cdot x \to 0 lim (\frac{1 - cos ( x )}{sin ( x ) + x cos ( x )})

使用洛必达法则

= 7 \cdot x \to 0 lim (\frac{sin ( x )}{2 cos ( x ) - x sin ( x )})

代入值

x = 0

= 7 \cdot \frac{sin ( 0 )}{2 cos ( 0 ) - 0 \cdot sin ( 0 )}

化简

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

= 0

作图

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