JEE Math Practice Question

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

1. A.$$\frac{5}{2}+\frac{\pi}{8}$$
2. B.$$\frac{3}{8}+\frac{\pi}{4}$$
3. C.$$\frac{3}{4}+\frac{\pi}{8}$$
4. D.$$\frac{3}{2}+\frac{\pi}{4}$$