$\lim _{x}$ ${\rightarrow 1} \frac{(1-x)(1-x^2) \cdots (1-x^{2n})}{\{(1-x)(1-x^2) \cdots (1-x^n)\}^2} = \dots, \forall n \in N$

$\mathop {\lim }\limits_{x \to 3} [x] = $,(where $[.]$ denotes the greatest integer function)

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

Evaluate $\mathop {\lim }\limits_{x \to 0} \frac{{{y^2}}}{x}$,where ${y^2} = ax + b{x^2} + c{x^3}$.

$\mathop {\lim }\limits_{x \to 1} f(x)$ is equal to,where $f(x) = \begin{cases} \frac{e^{\frac{1}{x-1}} - 2}{e^{\frac{1}{x-1}} + 2} & x \neq 1 \\ 1 & x = 1 \end{cases}$

If $f(x) = \begin{cases} \frac{\sin(1+[x])}{[x]}, & \text{for } [x] \neq 0 \\ 0, & \text{for } [x] = 0 \end{cases}$ where $[x]$ denotes the greatest integer function,then $\lim_{x \rightarrow 0^{-}} f(x)$ is equal to