How to remove the smallest term from asymptotic expansion?

It is well-known that $e^{-1/x}sim o(x^n)$ as $xto 0^+$ for any $ninmathbb{N}$, thus if I do an asymptotic expansion for a function, say $f=1/(1-x)+e^{-1/x}$ as $xto 0^+$, I expect to receive an answer like $fsim 1+x + x^2+o(x^2)$. Nevertheless both Asymptotic and Series give me results including $e^{-1/x}$, see

In[1]:= Asymptotic[1/(1 – x) + E^(-1/x), {x, 0, 2}]
Out[1]:= 1 + E^(-1/x) + x + x^2
In[2]:= Series[1/(1 – x) + E^(-1/x), {x, 0, 2}]
Out[2]:= E^-(1/x)+O[x]^3+(1+x+x^2+O[x]^3)

My question is how to remove $e^{-1/x}$ in practice.

Thanks in advance!