Answer
\[f(x)=\frac{1}{x}\;,\; a=\frac{1}{4}\]
and \[\lim_{x\rightarrow 1/4}\frac{\displaystyle\frac{1}{x}-4}{x-\displaystyle\frac{1}{4}}=-16\]
Work Step by Step
Let \[L=\lim_{x\rightarrow 1/4}\frac{\displaystyle\frac{1}{x}-4}{x-\displaystyle\frac{1}{4}}\]
Compare $L$ with definition of derivative:
\[f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}\]
\[\Rightarrow f(x)=\frac{1}{x}\;,\; a=\frac{1}{4}\]
and \[L=f'(\frac{1}{4})\;\;\;\;\;\;\ldots (1)\]
\[f(x)=\frac{1}{x}\Rightarrow f'(x)=\frac{-1}{x^2}\]
\[\Rightarrow f'(\frac{1}{4})=-16\]
From (1)
\[L=-16\]
