Answer
a) The limit $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}$ is 0.
b) The aging rate of an astronaut traveling at speeds approaching the speed of light relative to the aging rate of a friend on Earth approaches 0, because from part (a), $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=0$. This means the astronaut will barely grow older.
c) The reason for using the left hand limit in part (a), $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}$ is that the speed of the astronaut cannot exceed the speed of light.
Work Step by Step
(a)
Consider the provided limit, $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}$.
Substitute ${{R}_{a}}={{R}_{f}}\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}}$
$\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}}$
Use limit property $\underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)\cdot g\left( x \right) \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)\cdot \underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$:
$\begin{align}
& \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}} \\
& =\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}}
\end{align}$
Use limit property $\underset{x\to a}{\mathop{\lim }}\,\sqrt{f\left( x \right)}=\sqrt{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}$ in $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}}$:
$\begin{align}
& \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\sqrt{1-\frac{{{v}^{2}}}{{{c}^{2}}}} \\
& =\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\left( 1-\frac{{{v}^{2}}}{{{c}^{2}}} \right)}
\end{align}$
Use limit property $\underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)-g\left( x \right) \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)-\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$ in $\sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\left( 1-\frac{{{v}^{2}}}{{{c}^{2}}} \right)}$:
$\begin{align}
& \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\left( 1-\frac{{{v}^{2}}}{{{c}^{2}}} \right)} \\
& =\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,1-\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\frac{{{v}^{2}}}{{{c}^{2}}}}
\end{align}$
Use quotient property of limits $\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)},\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)\ne 0$ in $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\frac{{{v}^{2}}}{{{c}^{2}}}$:
$\begin{align}
& \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,1-\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,\frac{{{v}^{2}}}{{{c}^{2}}}} \\
& =\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,1-\frac{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{v}^{2}}}{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{c}^{2}}}}
\end{align}$
Use limit property $\underset{x\to a}{\mathop{\lim }}\,c=c $
$\begin{align}
& \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{f}}\cdot \sqrt{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,1-\frac{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{v}^{2}}}{\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{c}^{2}}}} \\
& ={{R}_{f}}\cdot \sqrt{1-\frac{{{c}^{2}}}{{{c}^{2}}}}
\end{align}$
Simplify it further:
$\begin{align}
& \underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}={{R}_{f}}\cdot \sqrt{1-1} \\
& ={{R}_{f}}\cdot \sqrt{0} \\
& ={{R}_{f}}\cdot 0 \\
& =0
\end{align}$
Thus the limit $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}$ is 0.
(b)
From part (a), $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}=0$
This means that as the speed $ v $ approaches the speed of light, $ c $, the aging rate of the astronaut relative to the aging rate of the friend on Earth approaches 0.
(c)
The left hand limit is used in $\underset{v\to {{c}^{-}}}{\mathop{\lim }}\,{{R}_{a}}$ as it is not possible physically to exceed the speed of light, $ c $.
So, the velocity $ v $ by which the astronaut is moving is approaching the speed of light, $ c $ but is less than $ c $.
