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受欢迎的微积分 >

limit as x approaches 0+of (1-sqrt(cos(x)))/(x(1-cos(sqrt(x))))

解答

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

解答

\frac{1}{2}

+1

十进制

0.5

求解步骤

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

乘开

x (1 - cos (x)) : x - x cos (x)

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

使用洛必达法则

= x \to 0 + lim \frac{\frac{s i n ( x )}{2 c o s ( x )}}{1 - cos ( x ) + \frac{x s i n ( x )}{2}}

化简

\frac{\frac{s i n ( x )}{2 c o s ( x )}}{1 - cos ( x ) + \frac{x s i n ( x )}{2}} : \frac{sin ( x )}{cos ( x ) ( 2 - 2 cos ( x ) + x sin ( x ) )}

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

使用洛必达法则

= x \to 0 + lim \frac{cos ( x )}{- \frac{s i n ( x ) ( 2 - 2 c o s ( x ) + x s i n ( x ) )}{2 c o s ( x )} + \frac{c o s ( x ) ( 3 s i n ( x ) + x c o s ( x ) )}{2 x}}

化简

\frac{cos ( x )}{- \frac{s i n ( x ) ( 2 - 2 c o s ( x ) + x s i n ( x ) )}{2 c o s ( x )} + \frac{c o s ( x ) ( 3 s i n ( x ) + x c o s ( x ) )}{2 x}} : \frac{2 x cos ( x ) cos ( x )}{- x sin ( x ) ( 2 - 2 cos ( x ) + x sin ( x ) ) + cos ( x ) ( 3 sin ( x ) + x cos ( x ) )}

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

使用洛必达法则

= x \to 0 + lim \frac{2 ( \frac{1}{2 x} cos ^{\frac{3}{2}} ( x ) + ( - \frac{3 s i n ( x ) c o s ( x )}{2} ) x )}{\frac{- 3 x s i n ( x ) c o s ( x ) - 4 x c o s ( x ) + 4 x c o s ( x ) c o s ( x ) - 2 x x c o s ( x ) s i n ( x ) - 10 x s i n ( x ) s i n ( x ) - x s i n ( x ) c o s ( x ) - 2 s i n ( x ) + 2 s i n ( x ) c o s ( x ) + 4 c o s ( x ) c o s ( x )}{2 x}}

化简

\frac{2 ( \frac{1}{2 x} cos ^{\frac{3}{2}} ( x ) + ( - \frac{3 s i n ( x ) c o s ( x )}{2} ) x )}{\frac{- 3 x s i n ( x ) c o s ( x ) - 4 x c o s ( x ) + 4 x c o s ( x ) c o s ( x ) - 2 x x c o s ( x ) s i n ( x ) - 10 x s i n ( x ) s i n ( x ) - x s i n ( x ) c o s ( x ) - 2 s i n ( x ) + 2 s i n ( x ) c o s ( x ) + 4 c o s ( x ) c o s ( x )}{2 x}} : \frac{2 ( - 3 x sin ( x ) cos ( x ) + cos ^{\frac{3}{2}} ( x ) )}{- 3 x sin ( x ) cos ( x ) - 4 x cos ( x ) + 4 x cos ( x ) cos ( x ) - 2 x x cos ( x ) sin ( x ) - 10 x sin ( x ) sin ( x ) - x sin ( x ) cos ( x ) - 2 sin ( x ) + 2 sin ( x ) cos ( x ) + 4 cos ( x ) cos ( x )}

= x \to 0 + lim \frac{2 ( - 3 x sin ( x ) cos ( x ) + cos ^{\frac{3}{2}} ( x ) )}{- 3 x sin ( x ) cos ( x ) - 4 x cos ( x ) + 4 x cos ( x ) cos ( x ) - 2 x x cos ( x ) sin ( x ) - 10 x sin ( x ) sin ( x ) - x sin ( x ) cos ( x ) - 2 sin ( x ) + 2 sin ( x ) cos ( x ) + 4 cos ( x ) cos ( x )}

代入值

x = 0

= \frac{2 ( - 3 \cdot 0 \cdot sin ( 0 ) cos ( 0 ) + cos ^{\frac{3}{2}} ( 0 ) )}{- 3 \cdot 0 \cdot sin ( 0 ) cos ( 0 ) - 4 \cdot 0 \cdot cos ( 0 ) + 4 \cdot 0 \cdot cos ( 0 ) cos ( 0 ) - 2 \cdot 0 \cdot 0 cos ( 0 ) sin ( 0 ) - 10 0 sin ( 0 ) sin ( 0 ) - 0 sin ( 0 ) cos ( 0 ) - 2 sin ( 0 ) + 2 sin ( 0 ) cos ( 0 ) + 4 cos ( 0 ) cos ( 0 )}

化简

\frac{2 ( - 3 \cdot 0 \cdot sin ( 0 ) cos ( 0 ) + cos ^{\frac{3}{2}} ( 0 ) )}{- 3 \cdot 0 \cdot sin ( 0 ) cos ( 0 ) - 4 \cdot 0 \cdot cos ( 0 ) + 4 \cdot 0 \cdot cos ( 0 ) cos ( 0 ) - 2 \cdot 0 \cdot 0 cos ( 0 ) sin ( 0 ) - 10 0 sin ( 0 ) sin ( 0 ) - 0 sin ( 0 ) cos ( 0 ) - 2 sin ( 0 ) + 2 sin ( 0 ) cos ( 0 ) + 4 cos ( 0 ) cos ( 0 )} : \frac{1}{2}

= \frac{1}{2}

作图

流行的例子

limit as x approaches 0 of+(x^{sqrt(x)})

x \to 0 lim (+ (x^{x}))

limit as x approaches 0 of ln(sqrt(x)+1)

x \to 0 lim (ln (x + 1))

limit as x approaches 2-of x/(x+2)

x \to 2 - lim (\frac{x}{x + 2})

limit as x approaches 0 of arctan(2/x)

x \to 0 lim (arctan (\frac{2}{x}))

limit as x approaches 4 of (x^2-1)/(x-1)

x \to 4 lim (\frac{x ^{2} - 1}{x - 1})