\(\dfrac{lim}{x\rightarrow1}\dfrac{\sqrt[2020]{2x-1}-1}{x-1}\)
\(=\dfrac{lim}{x\rightarrow1}\dfrac{2x-1-1}{\left(x-1\right)\left[\sqrt[2020]{\left(2x-1\right)^{2019}}+\sqrt[2020]{\left(2x-1\right)^{2018}}+...+\sqrt[2020]{2x-1}+1\right]}\)
\(=\dfrac{lim}{x\rightarrow1}\dfrac{2\left(x-1\right)}{\left(x-1\right)\left[\sqrt[2020]{\left(2x-1\right)^{2019}}+\sqrt[2020]{\left(2x-1\right)^{2018}}+...+\sqrt[2020]{2x-1}+1\right]}\)
\(=\dfrac{lim}{x\rightarrow1}\dfrac{2}{\sqrt[2020]{\left(2x-1\right)^{2019}}+\sqrt[2020]{\left(2x-1\right)^{2018}}+...+\sqrt[2020]{2x-1}+1}\)
\(=\dfrac{2}{1+1+1+...+1+1}=\dfrac{2}{2020}=\dfrac{1}{1010}\)
\(lim\dfrac{\sqrt[2020]{2x-1}-1}{x-1}=lim\dfrac{x\left(\sqrt[2020]{\dfrac{2}{x^{2019}}-\dfrac{1}{x^{2020}}}-\dfrac{1}{x^{2020}}\right)}{x\left(1-\dfrac{1}{x}\right)}\)
\(=\dfrac{0}{1}=0\)
