$f$is a continuous function in $[a, b]$ and $g$ is a continuous function in $[b, c] .$ A function $h\left(x\right)$ is defined as $h\left(x\right)=\left\{\begin{array}{ll}f\left(x\right),& x\in [a, b)\\ g\left(x\right),& x\in (b, c]\end{array}\right.$
Now, if $f\left(b\right)=g\left(b\right)$, then which of the following is always true.

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