Question

lim\(\dfrac{2\sqrt{n}-sin\sqrt{n}}{3\sqrt{n+2}+5}\)

Answer 1

Ta có: \(-\dfrac{1}{\sqrt{n}}\le\dfrac{sin\sqrt{n}}{\sqrt{n}}\le\dfrac{1}{\sqrt{n}}\) , mà

 \(\lim\left(-\dfrac{1}{\sqrt{n}}\right)=\lim\left(\dfrac{1}{\sqrt{n}}\right)=0\Rightarrow\lim\left(\dfrac{sin\sqrt{n}}{\sqrt{n}}\right)=0\)

Do đó:

\(\lim\dfrac{2\sqrt{n}-sin\sqrt{n}}{3\sqrt{n+2}+5}=\lim\dfrac{2-\dfrac{sin\sqrt{n}}{\sqrt{n}}}{3\sqrt{1+\dfrac{2}{n}}+\dfrac{5}{\sqrt{n}}}=\dfrac{2-0}{3+0}=\dfrac{2}{3}\)