Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 52

According to theory of relativity, the Lorenz Contraction Formula $L = L_0 \sqrt{1 - \frac{v^2}{c^2}}$ expresses
the length $L$ of an object as a function of its velocity $v$ with respect to an observer, where $L_0$ is the
length of the object at rest and $c$ is the speed of light. Find $\lim \limits_{v \to c^-} L$ and interpret
the result. Why is a left-hand limit necessary?





$
\begin{equation}
\begin{aligned}
\lim \limits_{v \to c^-} L & = \lim \limits_{v \to c^-} L_0 \sqrt{1 - \frac{v^2}{c^2}}\\
& = L_0 \sqrt{1 - \frac{c^2}{c^2}}\\
& = L_0 \sqrt{1 - 1} \\
& = 0
\end{aligned}
\end{equation}
$


As the velocity $v$ with respect to the observer approaches the speed of light $c$, the length $L$ of an object approaches .
The left hand limit is necessary because the function is defined only for $c > v$ because of the square root function.