Limits, L'Hôpital's rule, and epsilon delta definitions | Chapter 7, Essence of calculus

Limits sit at the center of calculus not as a new intuition, but as the rigorous language that makes “approach” precise—especially when derivatives are defined as limits. The derivative at a point is framed as the ratio of output change to input change, then interpreted through a limiting process: start with a small, ordinary nudge h away from the input, compute the resulting change in f, and ask what the ratio does as h shrinks toward 0. Crucially, this formal setup avoids talk of infinitesimals. The symbols dx and df can be treated as concrete finite nudges, provided the analysis explicitly tracks what happens as the nudge size approaches 0.

That emphasis on “approaches” leads directly to the epsilon-delta definition of limits, which turns the vague idea of getting closer into a testable condition. Using the example (2+h^3−2^3)/h, the expression is undefined at h=0 because it becomes 0/0, but it is well-defined for h values arbitrarily close to 0. As h approaches 0, the outputs narrow toward 12 from both sides, so the limit exists and equals 12. The mathematician’s skeptical question—what exactly does “approach” mean—gets answered by looking at output ranges. A limit exists when, for every tolerance ε (no matter how small), one can choose an input window δ around the target point such that every input within δ (excluding the problematic point itself) produces outputs within ε of the limiting value.

When the output refuses to squeeze into any arbitrarily small neighborhood, the limit fails to exist. The transcript contrasts the smooth “approach to 12” behavior with a jump-like counterexample: approaching from the right yields 2, while approaching from the left yields 1. No matter how tiny the input range becomes, the corresponding outputs still straddle at least a width of 1, so there is no single number that the function approaches. Epsilon-delta formalizes that failure: there exists some ε for which no δ can force the outputs to stay within ε of a candidate limit.

After building the rigor, the discussion shifts to computation. Many limits produce indeterminate forms like 0/0, such as sin(πx)/(x^2−1) near x=1 (and similarly near x=−1). Since direct substitution fails, the method uses a derivative-based approximation: when two functions f(x) and g(x both vanish at x=a), their values near a behave like their derivatives times the small nudge (roughly f′(a)·dx and g′(a)·dx). In the ratio f(x)/g(x), the dx factors cancel, leaving the limit equal to f′(a)/g′(a). This is L’Hôpital’s Rule, presented as a systematic way to turn certain indeterminate limits into derivative ratios. The rule is attributed historically to Johann Bernoulli, with L’Hôpital later associated with the work through patronage.

Overall, the through-line is that limits make calculus precise: they justify derivative definitions without infinitesimals, define what “approach” means in a way that can succeed or fail, and power practical tools like L’Hôpital’s Rule for evaluating otherwise stubborn 0/0 limits.