Part I — Limits and Continuity

"The whole of mathematical analysis is constructed in the realm of 'the very small.'" — Augustin-Louis Cauchy, 1821

This first part of the book builds the conceptual foundation on which everything else rests. We will not yet learn to differentiate or integrate. We will not yet compute derivatives or evaluate integrals. We will instead learn to think the way calculus requires you to think: in terms of limits — what happens when something gets infinitely close to something else without ever quite reaching it.

The limit is the central idea of calculus. Newton and Leibniz, working independently in the 1660s and 1670s, both used limits implicitly to define the new mathematics they were creating. But it took another two centuries — and the work of Cauchy, Weierstrass, and others — to make the limit concept rigorous. Today we begin with the intuition Newton had and end with the precision Weierstrass demanded.

What This Part Covers

  • Chapter 1 — Why Calculus. Two ancient problems (the tangent and the area) that drove mathematicians for two millennia. Why neither could be solved with pre-calculus mathematics. The astonishing realization that they are the same problem. A roadmap of where this book takes you.

  • Chapter 2 — Functions and Models. A review of the functions you already know and the modeling viewpoint we will adopt throughout the book. Python's matplotlib is introduced gently as a tool for visualizing functions.

  • Chapter 3 — The Limit. The most important concept in calculus. We define limits intuitively, develop computational rules, meet the squeeze theorem, and end with the formal $\varepsilon$-$\delta$ definition that Cauchy and Weierstrass spent a century perfecting.

  • Chapter 4 — Continuity. When a function behaves nicely under limits. The Intermediate Value Theorem, proved and applied. Why continuity is the unspoken assumption behind almost every theorem in calculus.

  • Chapter 5 — Rates of Change. Average rate vs. instantaneous rate. The tangent line and the velocity problems. The derivative defined (as a limit) but not yet computed — the computation comes in Part II.

What You Should Be Able to Do by the End of Part I

  • Evaluate a wide variety of limits using algebraic manipulation, the squeeze theorem, and limit laws
  • Recognize where a function is continuous, where it is discontinuous, and what kind of discontinuity it has
  • Apply the Intermediate Value Theorem to prove existence of solutions
  • Set up the limit-of-the-difference-quotient definition of the derivative for any function
  • Write the formal $\varepsilon$-$\delta$ definition of a limit (and execute a simple $\varepsilon$-$\delta$ proof, if you are a math major)
  • Use Python with sympy to compute limits and verify hand computations

Why This Part Matters

Many students arrive at calculus expecting to do something — compute derivatives, find integrals, solve equations. Part I will frustrate that expectation. We are not yet computing. We are defining. We are building the conceptual machinery that will allow Part II's computations to make sense.

If you skip Part I or rush through it, you will end up doing what many calculus students do: executing procedures without understanding them. That is the failure mode this book exists to prevent. Read carefully. Do the exercises. The investment pays off enormously in everything that follows.

The reward, by the end of Chapter 5, is that you will understand what a derivative is before learning how to compute one. That understanding is what separates calculus students who succeed from calculus students who struggle.

Let's begin.

Chapters in This Part