Chapter 3 — Further Reading

Each entry is annotated with what to read and why. Cross-references to our two anchor textbooks are mapped in appendices/appendix-h-stewart-chapter-mapping.md (Stewart) and appendices/appendix-i-openstax-chapter-mapping.md (OpenStax).

Anchor Textbook Sections (read alongside this chapter)

Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage.

Our primary breadth-and-exercise benchmark. Read §2.2 (the intuitive limit), §2.3 (limit laws — Stewart's treatment mirrors our Section 3.6), §2.4 (the precise ε-δ definition, our Section 3.10), and §2.5 (continuity, a preview of Chapter 4). §2.6 covers limits at infinity (our Section 3.5). §1.4 ("The Tangent and Velocity Problems") is the natural companion to Case Study 2.

OpenStax — Strang, G., and Herman, E. (2016). Calculus, Volume 1. OpenStax (free).

Free and excellent. Read §2.2 (intuitive limit and the numerical-table method we use in Section 3.2), §2.3 (limit laws and the Squeeze Theorem, our Sections 3.6–3.7), §2.4 (continuity), and §2.5 (the formal ε-δ definition, our Section 3.10). Its worked ε-δ examples are gentler than Stewart's and pair well with our Math Major Sidebar.

On the History and Meaning of the Limit

Boyer, C. B. (1959). The History of the Calculus and Its Conceptual Development. Dover.

Chapter 4 traces "limit" from Newton's "ultimate ratios" through Cauchy's Cours d'Analyse (1821) to Weierstrass's ε-δ formulation (1860s). Essential for understanding why the formal definition of Section 3.10 looks the way it does — and why it took two centuries to find.

Strogatz, S. (2019). Infinite Powers. Houghton Mifflin Harcourt.

Chapters 1–4 tell the story of limits, instantaneous speed, and the taming of infinity in vivid popular prose. The single best non-technical companion to this chapter; pairs directly with Case Study 2 on instantaneous velocity.

On Rigorous Limits (Math-Major Track)

Abbott, S. (2015). Understanding Analysis (2nd ed.). Springer.

The friendliest rigorous introduction. Chapter 4 (functional limits and continuity) develops the ε-δ machinery of Section 3.10 with unusually clear motivation. Start here if Spivak or Rudin feels steep.

Spivak, M. (2008). Calculus (4th ed.). Publish or Perish.

Chapters 5–6 give the gold-standard undergraduate treatment of limits, with many ε-δ proofs of escalating difficulty. Where this chapter's Part I exercises (3.36–3.42) end, Spivak begins.

Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.

"Baby Rudin." Chapter 4 (continuity, built from limits) is the classic — terse, demanding, and the foundation nearly every analyst eventually reads.

Bressoud, D. M. (2007). A Radical Approach to Real Analysis (2nd ed.). MAA.

Develops analysis through its historical struggles, showing why the rigorous limit evolved. Excellent for the reader who wants motivation alongside rigor.

On the Squeeze Theorem and the Sine Limit (Section 3.7)

Stewart, §2.3 and §3.3. The standard unit-circle geometric proof that $\lim_{x\to0}\frac{\sin x}{x} = 1$ via comparison of two triangles and a sector — the argument we sketch in Section 3.7 and complete in Chapter 7.

3Blue1Brown — "Limits, L'Hôpital's Rule, and Epsilon-Delta Definitions" (Essence of Calculus, ch. 7). YouTube.

A superb visual exposition of the ε-δ box picture and the squeeze idea. Strongly recommended after a first read of Sections 3.7 and 3.10.

On Floating-Point and the Numerical Limit (Case Study 1)

Goldberg, D. (1991). "What every computer scientist should know about floating-point arithmetic." ACM Computing Surveys 23(1), 5–48.

The canonical reference on IEEE 754 and catastrophic cancellation — the phenomenon driving the V-shaped error curve in Case Study 1. Read sections 1–2.

Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM.

The definitive scholarly treatment of rounding error. Chapter 1 frames the forward/backward-error viewpoint used to explain the optimal step $h^\star \sim \sqrt{\varepsilon_{\text{mach}}}$.

On Instantaneous Rates and the Physics of Falling (Case Study 2)

Galileo Galilei (1638). Two New Sciences.

The "Third Day" derivation of the $s \propto t^2$ law for falling bodies — the very law worked in Case Study 2, written a half-century before calculus existed.

Halliday, Resnick, and Walker. Fundamentals of Physics. Wiley. Chapter 2.

A standard physics treatment of average vs. instantaneous velocity, parallel to our difference-quotient development. Useful for seeing the same limit from a physicist's angle.

Computational References

SymPy documentation — https://docs.sympy.org/latest/modules/calculus/index.html

Reference for sp.limit, including the directional third argument '+'/'-' used in the exercises and one-sided/at-infinity limits.

NumPy documentation — https://numpy.org/doc/

For the numerical-evidence tables (np.linspace, vectorized evaluation) used throughout Section 3.9 and the case studies.

A Note for Math Majors: This Chapter Is Your First Taste of Analysis

The ε-δ definition of Section 3.10 is the doorway to real analysis (typically a junior-year course). A standard analysis sequence spends roughly six weeks each on (1) the construction of $\mathbb{R}$ and limits of sequences, (2) functional limits, continuity, and differentiability, and (3) the Riemann integral and series. The ε-δ skeleton you practice in exercises 3.36–3.42 will be your home for two semesters — Abbott or Spivak now will make all of it easier later. In still more general settings (metric and topological spaces), the limit reappears via open sets and neighborhoods, but Weierstrass's ε-δ idea remains underneath. It is one of the great mathematical achievements of the 19th century, and it has not been improved upon.