Let $f:(-\infty,\infty)\setminus{0}\to\mathbb{R}$ be differentiable such that $f'(1)=\lim_{a\to\infty} a^{2}f!\left(\tfrac{1}{a}\right)$. Then $\displaystyle \lim_{a\to\infty}\left(\frac{a(a+1)}{2}\tan^{-1}!\frac{1}{a}+a^{2}-2\log_{e}a\right)$ is: