What is the difference between semi-relativistic and non-relativistic limit of an equation? In special relativity, we often take the semi-relativistic and non-relativistic limit of an equation. When the relative velocity is much smaller than the velocity of light, i.e., ${v/c}\ll1 $, we consider the motion to be non-relativistic and the Lorentz factor $\gamma=\dfrac{1}{\sqrt{1-(v/c)^2}}$ becomes unity. But I couldn't clearly understand the notion of semi-relativistic limit.
 A: I don't think it makes sense to talk about a "semi-relativistic limit."
In the non-relativistic limit, as you say, we have $v/c \ll 1$, and we can use binomial expansion on the Lorentz factor to say things like
$$
E = \gamma mc^2 = mc^2 \times \left(
1 + \frac12\frac{v^2}{c^2} + \cdots
\right)
\approx mc^2 + \frac12 mv^2
\tag{slow}
$$
In the ultra-relativistic limit, we have $\gamma \gg 1$, and we can say things like
$$
E^2 = (pc)^2 + (mc^2)^2 \approx (pc)^2
\tag{fast}
$$
But if you're not working in either of these limits (if, for instance, you're in the regime where $\gamma\simeq2$ or $\gamma\simeq 10$), the neither of these approximations is correct or useful, and you have to keep track of all the $\gamma$s rather than using one of the above tricks to make them go away.  I have probably referred to this region as "semi-relativistic" in order to help my collider-physics colleagues remember that electrons do have mass. But it doesn't make sense to refer to this intermediate range as a "limit." It's just relativity.
If you're a three-significant-figures person, you might think "semi-relativistic" rather than the fast or slow limit for the interval $1.01 < \gamma < 100$. Sometimes it's nice to think about $\beta\gamma = \gamma v/c$ over this intermediate range, since the behavior of that product doesn't run up against an asymptote like $\beta$ or $\gamma$ do.
