Answer
(a)$ |f(x)-2| \lt4 \delta$
(b) $\delta=0.0025$
(c) $ \lim_{x\to 2}f(x)= 7$
Work Step by Step
(a) We have to find a relation between $|f(x)-2|$ and $|x-2| .$ Let $0\lt|x-2|\lt\delta$
\begin{aligned}
|f(x)-2| &=|4 x-1-7| \\
&=|4 x-8| \\
&=|4(x-2)| \\
&\lt4 \delta
\end{aligned}
(b) In order for $|(f(x)-7)|$ to be less then 0.01, our $\delta$ has to satisfy the next equation
\begin{array}{c}
{4 \delta=0.01} \\
{\delta=\frac{0.01}{4}=0.0025}
\end{array}
and
$$|f(x)-7|<4 \delta=4 * 0.0025=0.01$$
(c) From part (b), it follows that
$$ \lim_{x\to 2}f(x)= 7$$
