Why must the limits from the left and right at a point be equal for a function to be differentiable at that point?

Differentiation of a function (f) produces a new function (f'), which at any particular point (x), will take on the value of the slope of the initial function at that point. If the left and right limits of f are not equal, then there is a discontinuity at point, and the slope cannot be expressed, this means that the derivative will not be defined at this point, and differentiation is not valid there.

What does the Lorentz Factor $$\gamma = 1 / \sqrt{1 - v^{2}/c^{2} }$$ in the Lorentz Transformation $$x'=\gamma (x-vt)$$ imply regarding velocity limits in the universe, and why does it imply this?

The Lorentz Factor implies that nothing can travel faster than the speed of light, this is because in the limit as velocity approaches the speed of light, the Factor approaches a singularity where it grows without bound $$1/\sqrt{1 - c^{2}/c^{2}}=1/\sqrt{1 - 1}=1/0$$ . The mathematics break down at that point, indicating a situation which is not physically real.

What is Newton's Second Law of Motion, how does it bridge the First and Third?

Newton's Second Law is F=ma This states that the force applied to a mass is directly proportional to the acceleration the mass undergoes due to that force. This law includes the concept of inertia, essentially Newton's First Law, in that the increasing the mass of an object can be seen as increasing the force necessary to reach a certain acceleration, or decreasing the acceleration which occurs due to the same force. The Second Law also implies the Third, equal and opposite reactions, since any object that is subject to a force, but does not accelerate (such as pushing against a wall), must have zero net force acting on it, which implies there is another force equal in magnitude, but opposite in direction, acting to cancel out the applied force.