The value of $\mathop {\lim }\limits_{n \to \infty } \frac{{{x^n}}}{{{x^n} + 1}}$ where $x < -1$ is

$\lim _{x \rightarrow 1} \frac{(2 x-3)(\sqrt{x}-1)}{2 x^2+x-3} = $

$\lim _{x \rightarrow 0} \frac{1-\cos x \cos 2 x}{\sin ^2 x} = $

The value of $\mathop {\lim }\limits_{n \to \infty } {\left( {e \cdot {a^2} \cdot {e^3} \cdot {a^4} \cdots {e^{n - 1}} \cdot {a^n}} \right)^{\frac{1}{{{n^2} + 1}}}}$ is equal to

$\lim _{n}$ ${\rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\ldots +\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\ldots +n^3\right)-\left(1^2+2^2+\ldots +n^2\right)}$ is equal to:

$\mathop {\lim }\limits_{x \to \infty } {\left[ {1 + \frac{1}{{mx}}} \right]^x}$ is equal to