What can be said about the quadratic equation $a{x}^2+bx+c=0$ if $2a+3b+6c=0,$ $\left(a\mathrm{,} b\mathrm{,} c\in R\right) ?$

For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function $f:R\to R$ be given by

$f\left(x\right)=\left\{\begin{array}{cll}{a}_n+\sin \pi x,& \text{ for }& x\in [2n,2n+1]\\ b_n+\cos \pi x,& \text{ for }& x\in (2n-1,2n)\end{array}\right.$ for all integers $n$.  If $f$ is continuous, then which of the following hold(s) for all $n$?

Given $f\left(x\right)=\left\{\begin{array}{ll}3-\left[{\cot }^{-1}\left(\frac{2x^3-3}{x^2}\right)\right]& \mathrm{for} x>0\\ \left\{x^2\right\}\cos \left(e^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)& \mathrm{for} x<0\end{array}\right.$, (where, $\{·\}$ and $[·]$ denotes the fractional part and the integral part functions respectively).

Then which of the following statements do/does not hold good?

A function $f:R\to R$ is defined as

 $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{a{x}^2+bx+c+e^{nx}}{1+c·e^{nx}}$, where $f$ is continuous on $R$, then