\[\begin{split} &\lim_{x \to \infty} \frac{\sqrt{4x^2+1}}{x-1}=\\[15pt] &=\lim_{x \to \infty} \frac{\sqrt{4x^2+1}}{x-1}\frac{:x}{:x}=\\[15pt] &=\lim_{x \to \infty}\frac{\sqrt{\dfrac{4x^2}{x^2}+\dfrac{1}{x^2}}}{\dfrac{x}{x}-\dfrac{1}{x}}=\\[15pt] &=\lim_{x \to \infty}\frac{\sqrt{4+\dfrac{1}{x^2}}}{1-\dfrac{1}{x}}=\\[15pt] &=\frac{\sqrt{4}}{1}=2 \end{split}\]