Question

\(\lim\limits_{x\rightarrow+\infty}\left(x+1\right)\left(\sqrt{\dfrac{x}{2x^4+x^2+1}}\right)\)

Answer 1

\(\lim\limits_{x\rightarrow+\infty}\left(x+1\right)\left(\sqrt{\dfrac{x}{2x^4+x^2+1}}\right)\)

\(=\lim\limits_{x\rightarrow+\infty}\left(x+1\right)\left(\dfrac{\sqrt{x}}{x^2\cdot\sqrt{2+\dfrac{1}{x^2}+\dfrac{1}{x^4}}}\right)\)

\(=\lim\limits_{x\rightarrow+\infty}\left(\dfrac{\left(x+1\right)\cdot\sqrt{x}}{x^2\cdot\sqrt{2+\dfrac{1}{x^2}+\dfrac{1}{x^4}}}\right)\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{x\sqrt{x}+\sqrt{x}}{x^2\cdot\sqrt{2+\dfrac{1}{x^2}+\dfrac{1}{x^4}}}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{1}{\sqrt{x}}+\dfrac{1}{x\sqrt{x}}}{\sqrt{2+\dfrac{1}{x^2}+\dfrac{1}{x^4}}}=\dfrac{0+0}{\sqrt{2+0+0}}=0\)