Answer
The solution is
$$\lim_{v\to3}\frac{v-1-\sqrt{v^2-5}}{v-3}=-\frac{1}{2}.$$
Work Step by Step
To solve this limit follow the steps below
$$\lim_{v\to3}\frac{v-1-\sqrt{v^2-5}}{v-3}=\lim_{v\to3}\frac{v-1-\sqrt{v^2-5}}{v-3}\cdot\frac{v-1+\sqrt{v^2-5}}{v-1+\sqrt{v^2-5}}=\lim_{v\to3}\frac{(v-1)^2-\sqrt{v^2-5}^2}{(v-3)(v-1+\sqrt{v^2-5})}=\lim_{v\to3}\frac{v^2+1-2v-v^2+5}{(v-3)(v-1+\sqrt{v^2-5})}=\lim_{v\to3}\frac{-2v+6}{(v-3)(v-1+\sqrt{v^2-5})}=\lim_{v\to3}\frac{-2(v-3)}{(v-3)(v-1+\sqrt{v^2-5})}=\lim_{v\to3}\frac{-2}{v-1+\sqrt{v^2-5}}=\frac{-2}{3-1+\sqrt{3^2-5}}=\frac{-2}{2+2}=-\frac{1}{2}.$$
