As we have seen, can get the instantaneous
velocity by taking a limit
of the average velocity. If $s$ is a function of $t$,
then the instantaneous rate of change at $t=a$ is the limit of the
average velocity over shorter and shorter intervals. In other words,
we want to look at $$\lim_{t \to a} \frac{\Delta s}{\Delta t} =
\lim_{t \to a} \frac{s(t)-s(a)}{t-a}.$$

In the following video, we illustrate this concept
by considering the instantaneous velocity at time $t=2$ of a
particle with position $s(t)=4t^2+3$, where $t$ is time in
seconds, and $s$ is position in feet.

We now show here the computation of the
instantaneous velocity at time $t=2$ seconds of a particle with
position $s(t)=4t^2+3$ feet. We will need to compute
either $$(1) \lim_{t\to 2}\frac{s(t)-s(2)}{t-2},\qquad\text{ or
}\qquad (2) \lim_{h\to 0}\frac{s(2+h)-s(2)}{h}.$$

DO:Work
through these examples carefully. Try to compute both
limits before looking at the solutions below.
The first one is also done on the previous video.

Notice that in both computations (until the denominator was
cancelled) if we plugged $t=2$ in, we would get the
indeterminate form $\frac{0}{0}$ (try
it) so we had to do more work.

We see that the instantaneous
velocity at $t=2$ is 16 feet/second.
Notice that the units come from the units of the numerator
(feet) and the denominator (seconds) of the function we are
taking the limit of--the average velocity function).

Also notice that the work required in either
computation is similar. Often students prefer method (1),
but in the next section you will be forced to use method (2), so
be sure you practice it at least as much as method (1).