$\mathop {\lim }\limits_{x \to 0} \frac{x}{{\tan x}}$ is equal to

If $\mathop {\lim }\limits_{x \to 0} kx\,\text{cosec}\,x = \mathop {\lim }\limits_{x \to 0} x\,\text{cosec}\,kx$,then $k = $

The value of $ \lim _{\theta \rightarrow 0} \frac{1-\cos 4 \theta}{1-\cos 6 \theta} $ is

$\lim _{x \rightarrow 0^{-}} \frac{\sqrt{\frac{1}{2}(1-\cos ^2 x)}}{x}$ is equal to

The value of the limit $\lim_{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3x + \sin x)}{\left(2 \sin 2x \sin \frac{3x}{2} + \cos \frac{5x}{2}\right) - \left(\sqrt{2} + \sqrt{2} \cos 2x + \cos \frac{3x}{2}\right)}$ is

$\mathop {\lim }\limits_{x \to 2} \left( {\frac{{\sqrt {1 - \cos \{ 2(x - 2)\} } }}{{x - 2}}} \right) = $