How to formally prove, by Cauchy definition of limit, that $\lim\limits_{x\rightarrow 0^{+}}\frac{f(x)}{g(x)}=1$ implies that also $\lim\limits_{x\rightarrow 0^{+}}\frac{f^{-1}(x)}{g^{-1}(x)}=1$, where $f^{-1}$ (or $g^{-1}$) denotes the inverse function of $f$ (or $g$), functions $f$ and $g$ are both continuous and strictly increasing and $f(0)=g(0)=0$?
The limit related to inverse functions
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real-analysis