How to find an approximate values of rational function $f(x)$ for large $x$, neglecting $\frac{1}{x^4}$ and successive terms?

Question

This is the function that I want to find an approximate value for it neglecting $\displaystyle \frac{1}{x^4}$ and successive terms

$$ f(x)=\frac{25x}{(x-2)^2(x^2+1)}. $$

Answer 1

Write this with $t= 1/x$, to get: $$f(x)=\frac{25x}{(x-2)^2(x^2+1)}\to\frac{25 t^3}{(1-2 t)^2 \left(t^2+1\right)}= 25 t^3\cdot\frac{1}{(1-2 t)^2 \left(t^2+1\right)}$$ Now expand the fraction as a Taylor series to first order, to get: $$25t^3(1+4t)+O(t^5)$$ Or, back to $x$: $$\frac{25}{x^3}+\frac{100}{x^4}+O(x^{-5})$$