Question

\(\lim\limits_{x\rightarrow\infty}\left(\sin\left(\sqrt{x+1}\right)-\sin\sqrt{x}\right)\)

Answer 1

\(=\lim\limits_{x\rightarrow\infty}2cos\left(\dfrac{\sqrt{x+1}+\sqrt{x}}{2}\right)sin\left(\dfrac{\sqrt{x+1}-\sqrt{x}}{2}\right)\)

\(=\lim\limits_{x\rightarrow\infty}2cos\left(\dfrac{\sqrt{x+1}+\sqrt{x}}{2}\right)sin\left(\dfrac{1}{2\left(\sqrt{x+1}+\sqrt{x}\right)}\right)\)

Ta có:

\(-2\le2cos\left(\dfrac{\sqrt{x+1}+\sqrt{x}}{2}\right)\le2\) (hữu hạn)

\(\lim\limits_{x\rightarrow\infty}sin\left(\dfrac{1}{2\left(\sqrt{x+1}+\sqrt{x}\right)}\right)=sin\left(0\right)=0\)

\(\Rightarrow\lim\limits_{x\rightarrow\infty}\left(sin\sqrt{x+1}-sin\sqrt{x}\right)=0\)