Overview

The derivative — the engine of all of calculus — is defined as a limit. Without limits, differential calculus cannot begin. This topic develops the concept from two physical problems (tangent lines and instantaneous velocity), builds up the computational machinery (limit laws, algebraic techniques, the squeeze theorem), introduces the rigorous $\varepsilon$-$\delta$ definition, and establishes continuity — the hypothesis that makes the Intermediate Value Theorem and the Extreme Value Theorem work. Together these form the foundation on which differentiation and integration rest.

For quantitative finance, limits are not only foundational: they appear directly. The continuous compounding limit $\lim_{n\to\infty}(1+r/n)^n = e^r$ (Topic 1, Lesson 3) is a limit; the definition of a derivative (the sensitivity of an option price to a parameter) is a limit; the convergence of a binomial tree to the Black-Scholes price as the number of steps goes to infinity is a limit. Understanding how limits behave — and how they fail — is the only way to reason reliably about these convergences.

By the end of this topic you should be able to:

• state the informal limit definition and compute limits from tables and graphs;
• apply the sum, product, quotient, and power limit laws to evaluate limits algebraically;
• handle the $\varepsilon$-$\delta$ definition: interpret it, write a $\delta$-chase for a linear function, and verify a given limit claim;
• identify and classify all three types of discontinuity (removable, jump, infinite);
• state and apply the Intermediate Value Theorem to demonstrate the existence of a root;
• evaluate limits at infinity and identify horizontal asymptotes;
• compare the growth rates of polynomials, exponentials, and logarithms at $\pm\infty$.

1. Contents
2. The big picture