When performing a Series
expansion at infinity, Mathematica sometimes retains some exponential terms which for large expressions are making it impossible to understand the actual behaviour of a function in a given limit. As an example, consider the function
$$f(x)=x-\text{erf}(x).$$
Clearly one has $\lim_{x\rightarrow\infty}\left[f(x)-x\right]=-1$, so that for large $x$, one can say that
$$f(x)\approx x-1.$$
Now I would like to obtain such a result using Mathematica. Is there a specific way to do that?
One could think of using
Series[x-Erf[x],{x,∞,1}] // Normal//Expand
but would obtain the result $$\frac{e^{-x^2}}{\sqrt{\pi } x}+x-1$$ Here clearly it is easy to realise the term $\frac{e^{-x^2}}{\sqrt{\pi } x}$ can be ignored for large $x$. However this might not be the case when considering a more complicated function with dozen of terms like this. Is anyone aware of a way to retain only the significant part of the expression or alternatively another way to obtain the behaviour of the function for large $x$?