Behaviour a function when input is small

Question

If I have the function:

$P(N)=\frac{P_0N^2}{A^2+N^2}$, with $P_0, A$ positive constants

For small $N$, am I right in thinking that because $A$ dominates $N$ we have that $P(N) \approx \frac{P_0N^2}{A^2}$

Answer 1

More precisely, we can write for $A^2>N^2$

$$\begin{align} P(N)&=P_0\left(\frac{N^2}{A^2+N^2}\right)\\\\ &=P_0\left(\frac{(N/A)^2}{1+(N/A)^2}\right)\\\\ &=P_0\frac{N^2}{A^2}\sum_{k=0}^\infty (-1)^k\left(\frac{N^2}{A^2}\right)^{k}\\\\ &=\frac{P_0N^2}{A^2}-P_0\frac{N^4}{A^4}+P_0\frac{N^6}{A^6}+O\left(\frac{N^8}{A^8}\right) \end{align}$$

Therefore, if we retain only the first term in the expansion we can formally write

$$P(N)\approx \frac{P_0N^2}{A^2}$$

where the approximation error is of order $(N/A)^4$.