Mechanics 1: Linear Kinematics and Calculus - 3D Math Primer for Graphics and Game Development - page 494

smaller and smaller values of ∆t, we converge to a particular value. In
the parlance of calculus, this value of 1 furlong/minute is a limiting value,
meaning that as we take smaller and smaller positive values for ∆t, the
result of our computation approaches 1, but does not cross it (or reach it
exactly).
Convergence arguments such as this are defined with rigor in calculus
by using a formalized tool known as a limit. The mathematical notation
for this is

x(t + ∆t) − x(t)
∆t

v(t) = lim
∆t→0

. (11.2)

The notation '→' is usually read as “approaches” or “goes to.” So the right
side of Equation (11.2) might be read as

“The limit of x(t + ∆t) − x(t)
∆t

as ∆t approaches zero,”

or

“The limit as ∆t approaches zero of x(t + ∆t) − x(t)
∆t

.”

In general, an expression of the form lim a→k [blah] is interpreted to mean
“The value that [blah] converges to, as a gets closer and closer to k.”
This is an important idea, as it defines what we mean by instantaneous
velocity.

Instantaneous velocity at a given time t may be interpreted as the average
velocity of an interval that contains t, in the limit as the duration of the
interval approaches zero.

We won't have much need to explore the full power of limits or get
bogged down in the finer points; that is the mathematical field of analysis,
and would take us a bit astray from our current, rather limited, 12 objectives.
We are glossing over some important details 13 so that we can focus on one
particular case, and that is the use of limits to define the derivative.
The derivative measures the rate of change of a function. Remember
that “function” is just a fancy word for any formula, computation, or proce-
dure that takes an input and produces an output. The derivative quantifies
the rate at which the output of the function will change in response to a
change to the input. If x denotes the value of a function at a specific time t,

12 No pun intended. Regretfully.
13 Including such things as continuity, limits taken from the left, from the right, etc.