Precalculus B — Polynomials & Rational Functions

A first taste of the limit

continues from lesson 2 — values defined earlier in the course stay live here

ELI5: here's the doorway to calculus. The function (x² − 1)/(x − 1) is 0/0 at x = 1 — a hole. But factor the top, (x − 1)(x + 1)/(x − 1) = x + 1, and as x sneaks up on 1 the value heads straight for 2. That "value it heads toward" is a limit — the whole engine of Calculus 1.

near_{one} = \frac{1.0001^{2} - 1}{1.0001 - 1}

2.0001

evaluate just shy of x = 1

✓ pass

\mathrm{abs}\left(near_{one} - 2\right) < 0.01

it heads toward 2, hole or no hole

Where next: Calculus 1A makes this precise — limits, continuity, and the derivative.