Question

\(\dfrac{lim}{x\rightarrow1}\dfrac{\sqrt{2x-1}\sqrt[3]{5x-4}-1}{1-\sqrt{x}}\)

Answer 1

\(=\lim\limits_{x\rightarrow1}\dfrac{\left(2x-1\right)^{\dfrac{1}{2}}.\left(5x-4\right)^{\dfrac{1}{3}}-1}{1-x^{\dfrac{1}{2}}}\)

Co : \(\left[\left(2x-1\right)^{\dfrac{1}{2}}.\left(5x-4\right)^{\dfrac{1}{3}}\right]'\) \(=\dfrac{1}{2}\left(2x-1\right)^{-\dfrac{1}{2}}.2.\left(5x-4\right)^{\dfrac{1}{3}}+\dfrac{1}{3}\left(5x-4\right)^{-\dfrac{2}{3}}.5.\left(2x-1\right)^{\dfrac{1}{2}}\)

\(\Rightarrow\lim\limits_{x\rightarrow1}...=\dfrac{\dfrac{1}{2}.2.1+\dfrac{1}{3}.1.5.1}{-\dfrac{1}{2}.1}=-\dfrac{16}{3}\)