Rates, Ratios & the Idea of Approaching a Value

Think about the two numbers on a car's dashboard. The odometer just counts up total distance traveled — it's a running total, nothing more. The speedometer is stranger: it claims to tell you how fast you're going right now, at this exact instant, even though "speed" is really just distance divided by time, and at a single instant zero time has passed.

Here's the trick: take a tiny window of time around that instant — a hundredth of a second, say — and compute distance ÷ time over just that window. It won't be perfectly exact, but it'll be close. Shrink the window further and it gets closer still. The speedometer reading is whatever number that ratio keeps getting nearer and nearer to as the window shrinks toward nothing — without the window ever actually reaching zero width, since dividing by zero isn't allowed. That "getting closer without arriving" is the whole idea.

A dropped ball has fallen s(t) = t² meters after t seconds. What's its average rate of fall between t = 1 and t = 1.1 seconds?

2.1 is already close to a suspiciously round number: 2. Shrink the interval further — say to t = 1 to t = 1.01 — and the ratio creeps even closer to 2, without ever needing the interval to vanish entirely.

Drag the slider to shrink h, the width of the interval starting at t = 1. Watch the secant line swing to lie almost flat against the curve, and watch the average rate creep toward 2 — the slider can get arbitrarily close to h = 0, but it stops just short, because a zero-width interval isn't a ratio anymore.

The pattern above has a name. An average rate of change is a ratio of a change in output to a change in input, measured over an interval of width h:

As h approaches zero, this ratio approaches a single fixed number — the value it's heading toward, even though h = 0 itself is off-limits. That approached value is written with a limit: