
\begin{enumerate}
\item $\begin{aligned}[t]
f(a+b)-f(a-b) & =\dfrac{1}{a+b}-\dfrac{1}{a-b}\\
& = \dfrac{a-b}{(a+b)(a-b)}-\dfrac{a+b}{(a+b)(a-b)}\\
& = \dfrac{(a-b)-(a+b)}{(a-b)(a+b)}\\
& = \dfrac{a-b-a-b}{a^2-b^2}\\
\Aboxed{f(a+b)-f(a-b) & = \dfrac{-2b}{a^2-b^2}}
\end{aligned}$

\item $\begin{aligned}[t]
\dfrac{1}{2}\big[ f(a+b)+f(a-b) \big] & =\dfrac{1}{2}\left[\dfrac{1}{a+b}+\dfrac{1}{a-b}\right]\\
& = \dfrac{1}{2}\left[\dfrac{a-b}{(a+b)(a-b)}+\dfrac{a+b}{(a+b)(a-b)}\right]\\
& = \dfrac{1}{2}\left[\dfrac{(a-b)+(a+b)}{(a-b)(a+b)}\right]\\
& = \dfrac{1}{2}\left[\dfrac{a-b+a+b}{a^2-b^2}\right]\\
\Aboxed{\dfrac{1}{2}\big[f(a+b)+f(a-b)\big] & = \dfrac{a}{a^2-b^2}}
\end{aligned}$
\end{enumerate}
