Limit's Dilemma

Since the limit is in the form of $\boxed{1^{\infty}}$

Now, Generalised formula for $1^{\infty}$ if $\boxed{\lim_{x\to a}f(x)=1}$ and $\boxed {\lim_{x\to a}\pm \infty}$ is

$\boxed {\large \lim_{x\to0}(f(x))^{g(x)}=e^{\lim_{x\to0}g(x)(f(x)-1)}}$

Considering $f(x)=1+a x+b x^2$ and $g(x)=\frac{2}{x}$

Applying above formula $\large \lim_{x\to0} (1+ax+bx^2)^{2/x} = e^{\lim_{x\to0}\frac{2}{x}(a x +b x^2)}$

According to the question , $\large e^{\lim_{x\to0}\frac{2}{x}(a x +b x^2)}=e^3$

Comparing the exponents, we get

$\large \implies {\lim_{x\to0}\frac{2}{x}(a x +b x^2)}=3$

$\large \implies \lim_{x\to0} 2 a+2 b x=3$ $\large \implies 2 a+2 b (0)=3$

$\large \implies 2 a=3$ $\large \implies \boxed {a=1.5,b\in \mathbb R}$