Let $f(x)=2021x^{2021}+x+1$, and compute the following limit:$$\lim_{x\to \infty} \frac{f^{-1}(2021x)-f^{-1}(x)}{\sqrt[2021] x}$$
My attempt: i want to use mean value theorem to $f^{-1}(x)$ then we have:$ \frac{2020x}{((2021)^2 (η_y)^{2021}+1)(\sqrt[2021] x)}$.And $f(η_y)=η_x$, $η_x∈(x,2021x) $. We know $η_x$ tend to 0 as x tend to $\infty$ so from the expression of $f$ we know $η_y$ is also.but we don't know the quotient of $η_y$ and $x$, so i think this way is fail. But i can't do it in other way.