# If $\lim\limits_{x\to 0}(1+ax)^{\Large\frac{b}{x}}=e^2\qquad(a,b\in N)$ then

$\begin{array}{1 1}(a)\;a=4,b=2&(b)\;a=8,b=4\\(c)\;a=16,b=8&(d)\;None\;of\;these\end{array}$

$\lim\limits_{x\to 0}(1+ax)^{\Large\frac{b}{x}}=\lim\limits_{x\to 0}\bigg[(1+ax)^{\Large\frac{1}{ax}}\bigg]^{ab}$
$\Rightarrow e^{ab}$
$\Rightarrow e^2$[Given]
$\Rightarrow ab=2$
Hence (d) is the correct answer.