If $\underset{x\to 0}{lim} \frac{e^{ax}-e^x-x}{x^2}=\frac{3}{2}$, then $a=$

$\underset{x\to 0}{lim} \frac{e^{ax}-e^x-x}{x^2}=\frac{3}{2}$

Numerator $=1+ax+\frac{a^2{x}^2}{2}+\dots -\left(1+x+\frac{x^2}{2}\dots \right)-x$

$=(a-2)x+\frac{\left(a^2-1\right)}{2}{x}^2+\dots$

Now, $\underset{x\to 0}{lim} \frac{e^{ax}-e^x-x}{x^2}=\frac{3}{2}$

$\begin{array}{l}\Rightarrow \underset{x\to 0}{lim} \frac{(a-2)x+\left(\frac{a^2-1}{2}\right)x^2+\dots }{x^2}=\frac{3}{2}\\ \Rightarrow \frac{a^2-1}{2}=\frac{3}{2}\Rightarrow a=2\end{array}$