Question

Biết \(\lim\limits_{x\rightarrow+\infty}\left(\sqrt{4x^2-3x+1}-\left(ã+b\right)\right)=0\)

Tính a-4b 

Answer 1

\(\lim\limits_{x\rightarrow+\infty}\left(\sqrt{4\text{x}^2-3\text{x}+1}-\left(ax+b\right)\right)\\ =\lim\limits_{x\rightarrow+\infty}\left(\dfrac{4\text{x}^2-3\text{x}+1-\left(ax+b\right)^2}{\sqrt{4\text{x}^2-3\text{x}+1}+\left(ax+b\right)}\right)\\= \lim\limits_{x\rightarrow+\infty}\left(\dfrac{\left(4-a\right)\text{x}^2-\left(3+2\text{a}b\right)\text{x}+1+b^2}{\sqrt{4\text{x}^2-3\text{x}+1}+\left(ax+b\right)}\right)=0\\ \Rightarrow\left\{{}\begin{matrix}4-a=0\\3+2\text{a}b=0\\2+a\ne0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=4\\b=-\dfrac{3}{8}\\\end{matrix}\right.\)