Question

cho \(\lim\limits_{x\rightarrow0}\left(\dfrac{x}{\sqrt[7]{x+1}\sqrt{x+4}-2}\right)=\dfrac{a}{b}\). tìm a,b biết a/b tối giản

Answer 1

\(\lim\limits_{x\rightarrow0}\dfrac{x}{\sqrt[7]{x+1}\left(\sqrt[]{x+4}-2\right)+2\left(\sqrt[7]{x+1}-1\right)}\)

\(=\lim\limits_{x\rightarrow0}\dfrac{x}{\dfrac{x\sqrt[7]{x+1}}{\sqrt[]{x+4}+2}+\dfrac{2x}{\sqrt[7]{\left(x+1\right)^6}+\sqrt[7]{\left(x+1\right)^5}+\sqrt[7]{\left(x+1\right)^4}+\sqrt[7]{\left(x+1\right)^3}+\sqrt[7]{\left(x+1\right)^2}+\sqrt[7]{x+1}+1}}\)

\(=\dfrac{1}{\dfrac{1}{2+2}+\dfrac{2}{1+1+1+1+1+1+1}}=\dfrac{28}{15}\)