Approximation $\frac{\mathrm{d} x}{x+\mathrm{d} x} \approx \frac{\mathrm{d}x}{x} $?

Question

Why is it possible to do the following approximation?

$$\frac{\mathrm{d} x}{x+\mathrm{d} x} \approx \frac{\mathrm{d}x}{x} $$

Why does that $\mathrm{d} x$ in the denominator "count" less than the one in the numerator?

Answer 1

The $dx$ in the denominator is dwarfed by the $x$. The difference between $\frac{0.000001}{1.000001}$ and $\frac{0.000001}{1}$ is miniscule.

Answer 2

This is only valid as long as $x\not\approx 0$.

We have $x+\mathrm dx\approx x$ because $\mathrm dx\approx 0$ and addition is continuous. Then provided $x\not\approx 0$, the claim follows by continutity of division.

Answer 3

It's $\displaystyle \frac{dx}{x+dx}=\sum\limits_{k=0}^\infty (-1)^k (\frac{dx}{x})^{k+1}$.

Because of $\displaystyle (\frac{dx}{x})^{k+1}<<\frac{dx}{x}$ for $k\in\mathbb{N}$ you can approximate $\displaystyle \frac{dx}{x+dx}\approx \frac{dx}{x}$.

Answer 4

You are allowed to approx when terms are added as the difference between the results will be negligible however you cannot do the same with multiplication or division $10000+0.0001=10000.0001$ which is quite close however $10000*0.0001$ is definitely nowhere near $10000 $