Limits
Before the derivative and the integral comes the one idea that powers both: what value does f(x) head toward as x closes in on a point — even if the function never actually gets there? For Grade 12.
Getting close is the whole point
A limit doesn't ask "what is f at this point?" It asks a sneakier question: "as x sneaks up on c, where is f going?" You approach from the left, you approach from the right, and if both sides agree on a single height, that height is the limit — written
The twist: the function might have a hole at c, or jump, or shoot off to infinity — and the limit can still exist (or not) regardless of what happens exactly at c. Getting arbitrarily close is all that matters.
Sneak up on c 🎯
Slide the gap toward 0 (or hit "Approach c") and read the table: x marches toward c = 1 from both sides. Try each function — a hole, a jump, a smooth curve, and one that blows up.
Four things that can happen
The limit exists
On the removable hole and the continuous curve, both sides close in on the same number. There may be a gap in the graph — the limit doesn't care.
The limit fails
At a jump the left and right sides head to different heights, so there's no single answer. When it blows up, the values grow without bound — no finite limit.
Notice how the removable hole makes the point: (x²−1)/(x−1) is undefined at x = 1 (you'd divide by zero), yet the graph is a perfectly ordinary line heading straight for y = 2. The limit is 2 even though f(1) doesn't exist. That gap between "where it's going" and "what it is" is exactly what calculus learns to exploit.