Calculate: \lim_{x arrow 0} ((1-sqrt(cos(x)))/(x * sin(x)))

Expression: $\lim_{x \rightarrow 0} \left(\frac{ 1-\sqrt{ \cos\left({x}\right) } }{ x \times \sin\left({x}\right) }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 1-\sqrt{ \cos\left({x}\right) } \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x \times \sin\left({x}\right) \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ \frac{ \sin\left({x}\right) }{ 2\sqrt{ \cos\left({x}\right) } } }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( x \times \sin\left({x}\right) \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ \frac{ \sin\left({x}\right) }{ 2\sqrt{ \cos\left({x}\right) } } }{ \sin\left({x}\right)+x \times \cos\left({x}\right) }\right)$

Simplify the complex fraction

$\lim_{x \rightarrow 0} \left(\frac{ \sin\left({x}\right) }{ 2\sqrt{ \cos\left({x}\right) }\left( \sin\left({x}\right)+x \times \cos\left({x}\right) \right) }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow 0} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( \sin\left({x}\right) \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 2\sqrt{ \cos\left({x}\right) }\left( \sin\left({x}\right)+x \times \cos\left({x}\right) \right) \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ \cos\left({x}\right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 2\sqrt{ \cos\left({x}\right) }\left( \sin\left({x}\right)+x \times \cos\left({x}\right) \right) \right) }\right)$

Find the derivative

$\lim_{x \rightarrow 0} \left(\frac{ \cos\left({x}\right) }{ \frac{ -2{\sin\left({x}\right)}^{2}+8{\cos\left({x}\right)}^{2}-3x \times \sin\left({2x}\right) }{ 2\sqrt{ \cos\left({x}\right) } } }\right)$

Simplify the complex fraction

$\lim_{x \rightarrow 0} \left(\frac{ 2\sqrt{ \cos\left({x}\right) }\cos\left({x}\right) }{ -2{\sin\left({x}\right)}^{2}+8{\cos\left({x}\right)}^{2}-3x \times \sin\left({2x}\right) }\right)$

Evaluate the limit

$\frac{ 2\sqrt{ \cos\left({0}\right) }\cos\left({0}\right) }{ -2{\sin\left({0}\right)}^{2}+8{\cos\left({0}\right)}^{2}-3 \times 0\sin\left({2 \times 0}\right) }$

Simplify the expression

$\begin{align*}&\frac{ 1 }{ 4 } \\&\begin{array} { l }0.25,& {2}^{-2}\end{array}\end{align*}$