Match the following:

In the following, $[\mathrm{x}]$ denotes the greatest integer less than or equal to x .

The function $f(x)=\left\{\begin{array}{ll}e^x+a x & , x<0 \\ b(x-1)^2 & , x \geq 0\end{array}\right.$ is differentiable at $x=0$. Then

$$ \text { A function } f(x)=\left\{\begin{array}{cl} \frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right. $$