Exam problem

$\begin{aligned} L & = \lim_{n \to \infty} n \left(\sqrt{a_n^2+3}-\sqrt{a_n^2-3}\right) \\ & = \lim_{n \to \infty} n \left(\sqrt{a_n^2+3}-\sqrt{a_n^2-3}\right) \times \frac {\sqrt{a_n^2+3}+\sqrt{a_n^2-3}}{\sqrt{a_n^2+3}+\sqrt{a_n^2-3}} \\ & = \lim_{n \to \infty} \frac {6n}{\sqrt{a_n^2+3}+\sqrt{a_n^2-3}} & \small \color{#3D99F6} \text{Note that }a_n = 2+5(n-1) = 5n-3 \\ & = \lim_{n \to \infty} \frac {6n}{\sqrt{25n^2-30n+12}+\sqrt{25n^2-30n+6}} & \small \color{#3D99F6} \text{Divide up and down by }n \\ & = \lim_{n \to \infty} \frac 6{\sqrt{25-\frac {30}n+\frac {12}{n^2}}+\sqrt{25-\frac {30}n+\frac 6{n^2}}} \\ & = \frac 6{\sqrt {25}+\sqrt{25}} \\ & = \boxed{\dfrac 35} \end{aligned}$