What is the limit as x approaches 0+of (1-sqrt(cos(x)))/(x(1-cos(sqrt(x)))) ?

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

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

\frac{1}{2}

Decimal

0.5

Solution steps

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

Expand

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

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

Apply L'Hopital's Rule

= 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}}

Simplify

\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 ) )})

Apply L'Hopital's Rule

= 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}}

Simplify

\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 ) )})

Apply L'Hopital's Rule

= 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}}

Simplify

\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 )}

Plug in the value

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 )}

Simplify

\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}

What is the limit as x approaches 0+of (1-sqrt(cos(x)))/(x(1-cos(sqrt(x)))) ?
The limit as x approaches 0+of (1-sqrt(cos(x)))/(x(1-cos(sqrt(x)))) is 1/2