Question

lim ( \(^4\sqrt{16x^4+3x+1}-\sqrt{4x^2+2}\)) 

x-> +∞

Answer 1

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{16x^4+3x+1-16x^4}{\sqrt[4]{\left(16x^4+3x+1\right)^3}+2x\sqrt[3]{\left(16x^4+3x+1\right)^2}+4x^2\sqrt[3]{16x^4+3x+1}+8x^3}+\lim\limits_{x\rightarrow+\infty}\dfrac{4x^2-4x^2-2}{2x+\sqrt{4x^2+2}}\)

\(=\lim\limits_{x\rightarrow+\infty}\dfrac{\dfrac{3x}{x^3}+\dfrac{1}{x^3}}{\dfrac{\sqrt[4]{\left(16x^4+3x+1\right)^3}}{x^3}+\dfrac{2x\sqrt[3]{\left(16x^4+3x+1\right)^2}}{x^3}+\dfrac{4x^2\sqrt[3]{16x^4+3x+1}}{x^3}+\dfrac{8x^3}{x^3}}+\lim\limits_{x\rightarrow+\infty}\dfrac{-\dfrac{2}{x}}{\dfrac{2x}{x}+\dfrac{\sqrt{4x^2+2}}{x}}=0\)