Epsilon-Delta Proof

Calculus 1 · Axiom Academy

Proving a limit straight from the rigorous formal definition Use the formal definition of a limit to prove the following statement. The function is linear, so this is the cleanest possible setting to see the – machinery in action. We start from what we WANT — — and work backwards to a bound on |x-a| . This is exploration, not the proof itself; it tells us which to choose. Now we discard the scratch-work and write the real argument: fix any , take , assume , and derive . You just built a complete – proof, the rigorous foundation under every limit in calculus. Epsilon is the tolerance: it's how close we demand f(x) to be to L , and the challenger gets to pick it. Delta is our response: for each we must produce a that forces f(x) within of L . Scratch first, then prove: work backwards from to discover , then write the argument forward. Linear shortcut: for f(x)=mx+b the relationship is just — here m=3 , so . Why factor: pulling out the slope turns |f(x)-L| into a constant times |x-a| , which is exactly the quantity controls. Because the works for every , the limit holds — no matter how tight the tolerance.

This is the written version of the interactive lesson above. See the full Calculus 1 course.