Example 1

For $x$ real (resp. complex) tending to $+\infty$ (resp. $\infty$), every polynomial $a_0 x^n + a_1 x^{n-1} + \ldots + a_n$ with coefficients in V, such that $a_0 \neq 0$, has principal part $a_0 x^n$ with respect to the scale $x^n$ (or any scale containing the $x^n$). It follows that every rational fraction $\frac{a_0 x^n + \cdots + a_m}{b_0 x^n + \cdots + b_n}$ with real or complex coefficients such that $a_0 b_0 \neq 0$ has principal part $\frac{a_0}{b_0} x^n$ with respect to the same scale.