Let [t] be the greatest integer less than or equal to t. Then the least value of $p\in \mathbb{N}$ for which $...

$\mathop {\lim }\limits_{x \to 0} {\mathop{\rm cosec}\nolimits} x\left( {\sqrt {2{{\cos }^2}x + 3\cos x} - \sqrt {{{\cos }^2}x + \sin x + 4} } \right)$...

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6f\left( {\frac{1}{x}} \right) = \frac{{35}}{{3x}} - \frac{5}{2}$ . If the $...

If $\lim _{x \rightarrow \infty}\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{x}{1+x}\right)\right)^x=\alpha$, then the value of $\frac{\log _{\mathrm{e}} \alpha}{1+\log _{\mathrm{e}} \alpha}$ equals :