Ep 1: Newton's Calculus Was 'Broken' for 150 Years | What Is a Limit?

Calculus is built on a single idea, and almost every textbook hands it to you on the first day without saying where it came from or why it has to exist. That idea is the limit. This video walks through what a limit actually is — first using the intuition that Cauchy formalized in 1821, then the formal epsilon–delta definition that Karl Weierstrass nailed down a generation later — with the history of why it took mathematicians almost two hundred years to get the foundations right. Three worked examples: a full epsilon-delta proof of a simple limit done from the definition; a textbook 0/0 indeterminate form solved by factoring; and the geometric squeeze-theorem proof that sine of x over x tends to one. Whether you're starting Calculus 1 or coming back to brush up before integral or multivariable calculus, this is the foundation everything else in the course will rest on. 📚 Continues in: → 'One-Sided Limits and Continuity' — when limits exist, and when they don't. 🎓 Best for: Calculus 1, 2, and 3 students, AP Calculus AB / BC, anyone reviewing before an exam, and self-learners who want a single coherent course instead of a pile of disconnected YouTube videos. — QED Realized — Clean explanations of the ideas that take a real lecture to land. 🔔 Subscribe so you don't miss the next one:    / @qedrealized   #Calculus #Math #Mathematics #STEM #Limits #APCalculus #Calculus1 — Tags (for search) — what is a limit, limits calculus, calculus limits explained, epsilon delta definition of a limit, formal definition of a limit, Cauchy limit, Weierstrass epsilon delta, history of calculus, Newton fluxions, Berkeley ghosts of departed quantities, squeeze theorem, sin x over x limit, limit of (x^2-9)/(x-3), indeterminate form 0/0, calculus 1, AP Calculus AB, AP Calculus BC, learn calculus, calculus tutorial, MIT calculus ⏱ Chapters 0:00 What is a Limit? 0:34 Why calculus needs a new idea 1:25 Newton and Leibniz invent calculus — and a controversy 2:32 Berkeley's complaint: ghosts of departed quantities 3:22 Cauchy, 1821: the intuitive limit 4:18 Weierstrass, 1860s: the formal definition 5:34 What the definition is really saying 6:15 The algebraic limit laws 7:17 An epsilon–delta proof in action 8:28 Worked example 1: a removable 0/0 9:19 Worked example 1, continued 10:01 Worked example 2: setting up sin(x)/x 10:57 The squeeze theorem at zero 12:11 When a limit doesn't exist 13:06 Why this matters 13:43 Recap 14:32 Thanks for watching