Reduce error when $x$ is much smaller than 1 I have the following function $$\frac{1}{1+2x}-\frac{1-x}{1+x} $$
How to find equivalent way to compute it but when $x$ is much smaller than 1? I assume the problem here is with $1+x$ since it probably would be equal to 1. I don't know if multiplying by $(1-x)$ would be helpful as it would be
$$ \frac{1-x}{1+x-2x^2}-\frac{(1-x)^2}{1-x^2} $$ so there's still term $1+x$.

If the therms in $x^2$ can be neglegted:
$$\frac{1}{1+2x}\approx\frac{1-4x^2}{1+2x}=1-2x$$
$$\frac{1-x}{1+x}\approx\frac{(1-x)(1-x^2)}{1+x}=(1-x)^2\rightarrow$$
$$f(x)\approx1-2x-1+2x-x^2=-x^2$$
If therm in $x^3$ can be neglegted:
$$\frac{1}{1+2x}\approx\frac{1-8x^3}{1+2x}=1-2x+4x^2$$
$$\frac{1}{1+x}\approx\frac{1-x^3}{1+x}=1-x+x^2$$
And so on...
