Teach & case
Help a near-peer reason through the idea — when you explain it, you understand it best — then read the real decision it comes from.
Help Vela with the limit
In a minute you'll be the teacher. Get ready to teach Vela: a limit is the value a function APPROACHES, not its value AT the point.
A limit asks what value f(x) approaches as x gets close to a point. A hole at the point doesn't stop the limit from existing.
What you've shown you can teach
A case · you decide
Vela and the speed-camera instant
Vela — a student contesting a speed-camera ticket that claims an exact speed "at the instant" of the flash
The ticket says Vela's car was doing 70 km/h "at the instant" the camera flashed. But speed is distance ÷ time — and over an instant, time is zero. Distance ÷ 0? That's undefined. So how can a speed "at an instant" even mean anything?
Vela computes average speed over shorter and shorter windows around the flash: over 2 seconds, 68.4; over 1 second, 69.2; over 0.1 s, 69.9; over 0.01 s, 69.99…
The exact instant itself is still 0 ÷ 0 — genuinely undefined. But the averages don't wander; they home in on a single number. That approached value is what "instantaneous speed" means.
At the exact instant, the formula is 0 ÷ 0. So is the instantaneous speed simply undefined?
Vela's friend graphs a different function with a literal HOLE at x = 2 (undefined there). Does it have a limit as x → 2?
Vela's argument isn't that the camera is wrong — it's that "speed at an instant" is a LIMIT, and the limit is a clean 70. That's Vela's whole thing: the value a function approaches, not the value it happens to hit.
When two ideas meet: 2 cases with two people
A case · you decide
Vela and Nayeli: a slope built from a limit
Vela & Nayeli — a student who thinks in limits, and a friend who wants the slope of a curve at one point
Nayeli wants the slope of y = x² at x = 3. A slope needs two points, and she only has one.
Vela takes a second point a distance h away. The secant slope is ((3 + h)² − 9) ÷ h, which simplifies to 6 + h.
At h = 1 the secant slope is 7. At h = 0.1 it is 6.1. At h = 0.001 it is 6.001.
What is the slope of y = x² exactly at x = 3?
Vela's limit is what makes Nayeli's slope exact.
A case · you decide
Leda and Nayeli: where the slope is zero
Leda & Nayeli — a student maximising an area, and a friend who reads what the slope says
Leda has 40 metres of fence for a rectangular garden and wants the largest area.
If one side is x metres, the other is 20 − x, so the area is A = x(20 − x).
Nayeli computes the rate of change: A′ = 20 − 2x. It is positive while the area is still growing and negative once it starts shrinking.
Which rectangle gives the largest area?
Leda asked for the most. Nayeli showed where the slope says "no more".