\(=\lim\limits_{x\rightarrow+\infty}\dfrac{3x}{x^2+2}-\lim\limits_{x\rightarrow+\infty}\dfrac{5\sin2x}{x^2+2}+\lim\limits_{x\rightarrow+\infty}\dfrac{\cos^2x}{x^2+2}\)
\(\lim\limits_{x\rightarrow+\infty}\dfrac{3x}{x^2+2}=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{3x}{x^2}}{\dfrac{x^2}{x^2}+\dfrac{2}{x^2}}=0\)
\(-1\le\sin2x\le1\Rightarrow\dfrac{-5}{x^2+2}\le\dfrac{5\sin2x}{x^2+2}\le\dfrac{5}{x^2+2}\)
\(\lim\limits_{x\rightarrow+\infty}-\dfrac{5}{x^2+2}=\lim\limits_{x\rightarrow+\infty}\dfrac{5}{x^2+2}=0\Rightarrow\lim\limits_{x\rightarrow+\infty}\dfrac{5\sin2x}{x^2+2}=0\)
\(0\le\cos^2x\le1\Rightarrow0\le\dfrac{\cos^2x}{x^2+2}\le\dfrac{1}{x^2+2}\)
\(\lim\limits_{x\rightarrow+\infty}\dfrac{1}{x^2+2}=0\Rightarrow\lim\limits_{x\rightarrow+\infty}\dfrac{\cos^2x}{x^2+2}=0\)
\(\Rightarrow\lim\limits_{x\rightarrow+\infty}\dfrac{3x-5\sin2x+\cos^2x}{x^2+2}=0\)
