Mathematics - Limits Question with Solution | TestHub

$\begin{array}{rl}& f:(-\infty ,\infty )-\{0\}\to R\\ & f^{\prime }(1)=\underset{a\to \infty }{\lim }{a}^{2}f\left(\frac{1}{a}\right)\\ & \underset{a\to \infty }{\lim }\frac{a(a+1)}{2}{\tan }^{-1}\left(\frac{1}{a}\right)+a^2-2\ln (a)\\ & \underset{a\to \infty }{\lim }{a}^2\left(\frac{\left(1+\frac{1}{a}\right)}{2}{\tan }^{-1}\left(\frac{1}{a}\right)+1-\frac{2}{a^2}\ln (a)\right)\\ & f(x)=\frac{1}{2}(1+x){\tan }^{-1}(x)+1-2x^2\ln (x)\end{array}$
$\begin{array}{rl}& {\mathrm{f}}^{\prime }(\mathrm{x})=\frac{1}{2}\left(\frac{1+\mathrm{x}}{1+{\mathrm{x}}^2}+{\tan }^{-1}(\mathrm{x})+4\mathrm{x}\ell (\mathrm{x})\right)+2\mathrm{x}\\ & {\mathrm{f}}^{\prime }(1)=\frac{1}{2}\left(1+\frac{\pi }{4}\right)+2\\ & {\mathrm{f}}^{\prime }(1)=\frac{5}{2}+\frac{\pi }{8}\end{array}$