Let \(f:(-\infty ,\infty )-{0}\to \mathrm{ℝ}\) be a differentiable function such that \(f^{\prime}(1)=\lim _{a \rightarr…

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

[JEE Main 2024, 6 Apr (Shift 1)]