Limits and Differentiation from First Principles

Calculus begins with a deceptively simple question: how do we measure the instantaneous rate at which a quantity changes? Speed on a car's dashboard, the slope of a hillside at a single point, the marginal cost of one more unit — all of these ask for the behavior of a function at a point rather than over an interval. The tool that makes this precise is the limit.

The idea of a limit. We write $\lim_{x \to a} f(x) = L$ to mean that $f(x)$ can be made as close to $L$ as we like by taking $x$ sufficiently close to $a$ (but not equal to $a$). The value the function actually takes at $a$ is irrelevant; what matters is the value the outputs approach. For a continuous function like a polynomial, the limit is found simply by substitution: $\lim_{x \to 2}(x^2 + 1) = 5$. The interesting cases arise