\(\displaystyle \lim _{x \rightarrow-3} \sqrt{1-5 x}=\sqrt{1-5(-3)}=\sqrt{16}=4 \)

Step 1:
$$
\begin{aligned}
& |\sqrt{1-5 x}-4|<0.5 \Rightarrow-0.5<\sqrt{1-5 x}-4<0.5 \Rightarrow 3.5<\sqrt{1-5 x}<4.5 \Rightarrow 12.25<1-5 x<20.25 \\
& \Rightarrow 11.25<-5 x<19.25 \Rightarrow-3.85<x<2.25 \text {. }
\end{aligned}
$$

Step 2: $|x-(-3)|<\delta \Rightarrow-\delta<x+3<\delta \Rightarrow-\delta-3<x<\delta-3$.
Then $-\delta-3=-3.85 \Rightarrow \delta=0.85$, or $\delta-3=-2.25 \Rightarrow 0.75$; thus $\delta=0.75$.