limit as x approaches 0 of (ln(\frac{1+x)/(1-x))-2x}{x-sin(x)}

解答

x \to 0 lim (\frac{ln ( \frac{1 + x}{1 - x} ) - 2 x}{x - sin ( x )})

4

求解步骤

x \to 0 lim (\frac{ln ( \frac{1 + x}{1 - x} ) - 2 x}{x - sin ( x )})

使用洛必达法则

= x \to 0 lim (\frac{\frac{2}{( x + 1 ) ( - x + 1 )} - 2}{1 - cos ( x )})

使用洛必达法则

= x \to 0 lim (\frac{\frac{4 x}{( x + 1 ) ^{2} ( - x + 1 ) ^{2}}}{sin ( x )})

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= x \to 0 lim (\frac{4 x}{sin ( x ) ( x + 1 ) ^{2} ( - x + 1 ) ^{2}})

使用洛必达法则

= x \to 0 lim (\frac{4}{cos ( x ) ( x + 1 ) ^{2} ( - x + 1 ) ^{2} + ( 4 x ^{3} - 4 x ) sin ( x )})

代入值

x = 0

= \frac{4}{cos ( 0 ) ( 0 + 1 ) ^{2} ( - 0 + 1 ) ^{2} + ( 4 \cdot 0 ^{3} - 4 \cdot 0 ) sin ( 0 )}

化简

\frac{4}{cos ( 0 ) ( 0 + 1 ) ^{2} ( - 0 + 1 ) ^{2} + ( 4 \cdot 0 ^{3} - 4 \cdot 0 ) sin ( 0 )} : 4

= 4

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\frac{d}{d x} (\frac{14 x + 8 x ^{2}}{( 7 + 8 x ) ^{2}})

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