Question

$\underset{x\to \infty }{Lt}\sqrt{x^2-x+1}+ax-b=1$ Find $a$ and $b.$

Answer 1

$\begin{array}{c}{lim}_{x\to \infty }\sqrt{x^2-x+1}+ax-b=1\\ {lim}_{x\to \infty }\sqrt{\frac{1}{t^2}-\frac{1}{t}+1}+\frac{a}{t}-b\\ {lim}_{x\to \infty }\frac{\sqrt{1-t+t^2}+a-bt}{t}\\ {lim}_{x\to \infty }\frac{\left(1-\frac{t}{2}+\frac{t^2}{2}\right)+a-bt}{t}=1\\ \Rightarrow a=-1\\ -\frac{1}{2}-b=1\\ \Rightarrow b=\frac{-3}{2}\end{array}$