Local estimates for $|(x+\epsilon)^{-1} - x^{-1}|$

Question

I am interested in a local pertubation bound for the reciprocal function. How can you estimate the difference

$|(x+\epsilon)^{-1} - x^{-1}|$

where $x > 0$ and $\epsilon > 0$ is small? Even when $x$ is large, it should be possible to give a small range for $\epsilon$ where e.g. local Lipschitz continuity holds.

Answer 1

It seems the following.

Consider a function $f(t)=t^{-1}$. By Lagrange Theorem, there exists a number $c\in (x,x+\epsilon)$ such that

$$f(x+\epsilon) – f(x)=f’(c)\epsilon.$$

Thus

$$\frac{\epsilon}{(x+\epsilon)^2}<|(x+\epsilon)^{-1} - x^{-1}|=| f’(c)\epsilon |=\left|\frac 1{c^2}\epsilon\right|<\frac{\epsilon}{x^2}.$$