Answer
$$\lim _{h \rightarrow a} \frac{\frac{1}{h}-\frac{1}{a}}{h-a} =-\frac{1}{a^2}$$
Work Step by Step
Given $$\lim _{h \rightarrow a} \frac{\frac{1}{h}-\frac{1}{a}}{h-a}$$
let $$ f(x) = \frac{\frac{1}{h}-\frac{1}{a}}{h-a} $$
Since, we have
$$ f(a)= \frac{\frac{1}{a}-\frac{1}{a}}{a-a}=\frac{0}{0}$$
So, transform algebraically and cancel
\begin{aligned}L&=\lim _{h \rightarrow a} \frac{\frac{1}{h}-\frac{1}{a}}{h-a}\\
&=\lim _{h \rightarrow a} \frac{\frac{a-h}{ah}}{h-a}\\
&=\lim _{h \rightarrow a} \frac{-(h-a)}{ah(h-a)}\\
&=\lim _{h \rightarrow a} \frac{-1}{ah}\\
&=-\frac{1}{a^2}\\
\end{aligned}
