Asymptotic behavior of $-gTt-gT^2e^{\frac{-t}{T}}$ for small $t$ I want to solve this using Taylor series expansion of $e^{f(x)}$

$$\begin{align}x=-gTt-gT^2e^{\frac{-t}{T}}+gT^2+x_0\end{align}$$
Show that for small values of t $(t\ll T)$, the equation for x is approximately $x\approx{x_0-\frac{1}{2}gt^2}$ , where T is a constant

Please help.
Using Taylor Series You get:

$e^{-t/T} \approx 1 - t/T + 0.5 t^2 / T^2$
Hint: $e^{-t/T} \approx 1 - t/T + 0.5 t^2 / T^2$