Speed from GPS: The Limit

GPS only gives your position every so often, so your phone estimates speed as distance over time between two fixes. Shrink the time gap and that average closes in on one number: your speed at that instant. That number is a limit.

How it works

Limits

In one sentence

$\lim_{x\to a}f(x) = L$ means $f(x)$ can be made as close to $L$ as we like by taking $x$ close enough to $a$ (but not equal); the derivative is the limit of difference quotients, $s'(a) = \lim_{h\to0}\frac{s(a+h) - s(a)}{h}$ .

General formula

{\htmlData{sym=0}{v(a)}} = \lim_{{\htmlData{sym=2}{h}}\to 0}\frac{{\htmlData{sym=1}{s}}({\htmlData{sym=3}{a}} + {\htmlData{sym=2}{h}}) - {\htmlData{sym=1}{s}}({\htmlData{sym=3}{a}})}{\htmlData{sym=2}{{h}}}

Average speed over a tiny interval barely differs from the speed at its start. The limit is the value those averages settle on as the interval vanishes.

What the symbols mean

$v(a)$
Instantaneous velocity
Your speed at exactly time a.
$s$
Position
Distance along the road at each time, from GPS.
$h$
Time gap
Seconds between the two GPS fixes.
$a$
Moment
The time you want the speed at.

The average velocity over $[a, a + h]$ is the slope of the secant line between the two positions.
As $h$ shrinks, the secant pivots toward one line, and its slope approaches one value.
That value is the limit, the instantaneous velocity $v(a)$ , and the line is the tangent.
You never set $h = 0$ (that would divide by zero); the limit is what the slopes approach.

Learn the math

Tangent Lines and Rates of ChangeCalc I · Lesson 4, Limits The LimitCalc I · Lesson 5, Limits The Definition of the DerivativeCalc I · Lesson 14, Derivatives