Let $\mathrm{f}:(-\infty, \infty)-\{0\} \rightarrow \mathrm{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a$ is equal to $\_\_\_\_$ .