How to find an approximate values of rational function $f(x)$ for large $x$, neglecting $\frac{1}{x^4}$ and successive terms? This is the function that I want to find an approximate value for it neglecting $\displaystyle \frac{1}{x^4}$ and successive terms 
$$
f(x)=\frac{25x}{(x-2)^2(x^2+1)}.
$$

Since $\frac1{1+x}=1+O(x)$, we get
$$
\begin{align}
\frac{25x}{(x-2)^2(x^2+1)}
&=\frac{25}{x^3(1-\frac2x)^2(1+\frac1{x^2})}\\[9pt]
&=\frac{25}{x^3}\left(1+O\left(\frac1x\right)\right)^2\left(1+O\left(\frac1{x^2}\right)\right)\\[6pt]
&=\frac{25}{x^3}\left(1+O\left(\frac1x\right)\right)\\[6pt]
&=\frac{25}{x^3}+O\left(\frac1{x^4}\right)
\end{align}
$$
