NIMCET 2018 — Mathematics PYQ

Concept:
The limit of indeterminate form $1^{\infty }$ is calculated as follows:  
$\underset{x\to a}{\lim }f(x{)}^{g(x)}=e^{{\lim }_{x\to a}(f(x)-1)g(x)}$  

Calculation:  

$\underset{x\to \infty }{\lim }{\left(1+\frac{a}{x}+\frac{b}{x^2}\right)}^{2x}$  

$=e^{{\lim }_{x\to \infty }\left(1+\frac{a}{x}+\frac{b}{x^2}-1\right)(2x)}$  

$\Rightarrow e^{2a}=e^2$  

$\Rightarrow 2a=2$  

$\Rightarrow a=1$ and $b$ can take any real value because any real number divided by infinity is zero.  

Hence, $a=1$ and $b\in \mathbb{R}$.