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.
Let $y=x^{1/2021}$, then $x\to\infty$ iff $y\to\infty$, so we consider the limit $$\lim_{y\to\infty}\frac{f^{-1}(2021y^{2021})-f^{-1}(y^{2021})}{y}.$$ Note that $f$ is strictly increasing, so $f^{-1}(2021y^{2021})<y$. Note that $2021y^{2021}-f(y-1)$ is a polynomial of even degree $2020$ whose leading term has a positive coefficient $2021^2>0$, so when $|y|$ is sufficiently large, we have $f(y-1)<2021y^{2021}$ and thus $y-1<f^{-1}(2021y^{2021})$. Hence $$0<\left|\frac{f^{-1}(2021y^{2021})-y}{y}\right|<\frac1{|y|}, \qquad |y|>>1.$$ Therefore $$\lim_{y\to\infty}\frac{f^{-1}(2021y^{2021})}{y}=1.$$ And so $$\lim_{y\to\infty}\frac{f^{-1}(y^{2021})}{y}=\lim_{y\to\infty}\frac{f^{-1}(2021(2021^{-1/2021}y^{2021}))}{y}=2021^{-1/2021}.$$ Finally, $$\lim_{y\to\infty}\frac{f^{-1}(2021y^{2021})-f^{-1}(y^{2021})}{y}=1-2021^{-1/2021}=1-\left(\frac1{2021}\right)^{1/2021}.$$