Question

Tính giới hạn: \(\lim\limits_{x\rightarrow3}\dfrac{\sqrt{x+1}-\sqrt[3]{x+5}}{x-3}\)

Answer 1

\(\lim\limits_{x\rightarrow3}\dfrac{\sqrt[]{x+1}-\sqrt[3]{x+5}}{x-3}=\lim\limits_{x\rightarrow3}\dfrac{\sqrt{x+1}-2+2-\sqrt[3]{x+5}}{x-3}\)

\(=\lim\limits_{x\rightarrow3}\dfrac{\dfrac{x-3}{\sqrt[]{x+1}+2}-\dfrac{x-3}{4+2\sqrt[3]{x+5}+\sqrt[3]{\left(x+5\right)^2}}}{x-3}\)

\(=\lim\limits_{x\rightarrow3}\left(\dfrac{1}{\sqrt[]{x+1}+2}-\dfrac{1}{4+2\sqrt[3]{x+5}+\sqrt[3]{\left(x+5\right)^2}}\right)\)

\(=\dfrac{1}{2+2}-\dfrac{1}{4+4+4}=\dfrac{1}{6}\)