\[
  \begin{aligned}
    \lim\limits_{x\to+\infty}\frac{a\sqrt{2x^2+3}+2017}{2x+2018} &= \lim\limits_{x\to+\infty}\frac{a\sqrt{x^2(2 + \frac{3}{x^2})}+2017}{2x+2018} \\
                                                                 &= \lim\limits_{x\to+\infty}\frac{ax(\sqrt{2+\frac{3}{x^2}}+\frac{2017}{x})}{x(2+\frac{2018}{x})} \\
                                                                 &= \frac{a\sqrt{2}}{2}
  \end{aligned}
\]
Mà $\lim\limits_{x\to+\infty}\frac{a\sqrt{2x^2+3}+2017}{2x+2018} = \frac{1}{2} \implies \frac{a\sqrt{2}}{2} = \frac{1}{2} \implies a = \frac{\sqrt{2}}{2}$
