If $\underset{x→0}{\lim}(1+ax+bx^2)^{2/x}=e^3$, then
Answer & explanation
Correct answer: option 3
$\underset{x→0}{\lim}(1+ax+bx^2)^{2/x}=\underset{x→0}{\lim}(1+ax+bx^2)^{\frac{1}{ax+bx^2}\frac{2(ax+bx^2)}{x}}$
$=e^{\underset{x→0}{\lim}\frac{2(ax+bx^2)}{x}}e^{2a}=e^3⇒a=\frac{3}{2}$, b any real number.
Hence (C) is the correct answer.