Proving lim(x→2) x² = 4

Intro to Proofs · Axiom Academy

Proving straight from the epsilon–delta definition of a limit Using the formal definition of a limit, prove that . That is, show that for every there exists a such that whenever , we have . Pin the output x^2 inside the green band by squeezing the input x into the orange interval . The width of the orange δ-interval is ours to choose. The proof finds a δ small enough that the parabola carries every x inside it up into the green ε-band — no matter how thin that band is. That is a complete ε–δ proof. The engine behind it generalizes to nearly every polynomial limit: Scratchwork drives the choice of δ: we worked backward from to discover how must depend on , then wrote the argument forward. Factor to expose |x - 2| : writing surfaces the factor that directly controls. Bound the loose factor: |x + 2| is unbounded on its own, so we pre-restrict to trap x in (1, 3) and conclude |x + 2| < 5 . The pattern: here M = 5 , giving — one that honors both the " " restriction and the " " requirement. Same recipe every time: factor, restrict to bound the leftover factor by some M , then take . Master this and the ε–δ definition stops being intimidating.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.