Summary

This chapter built the idea of a limit from the ground up and used it to define continuity.

A limit asks what value $f(x)$ approaches as $x$ is pushed arbitrarily close to a point $x_0$, without ever landing on $x_0$ itself. Approaching from below gives the left-hand limit; approaching from above gives the right-hand limit; the ordinary (two-sided) limit exists exactly when both one-sided limits exist and agree, and its value is that common value. Whether or not $f$ is even defined at $x_0$ plays no role in whether the limit exists there — the two questions are genuinely independent.

Limits combine algebraically in the expected way — sums, differences, products, scalar multiples, and (denominator permitting) quotients of functions with existing limits themselves have limits obtained by combining the pieces — and for polynomials in particular, the limit can always be found by direct substitution. When direct substitution instead produces a meaningless $0/0$ or $\infty/\infty$ pattern (an indeterminate form), the fix is always algebraic: factor and cancel, rationalise, or divide through by a dominant power — never a numerical shortcut.

Sometimes $f(x)$ grows without bound rather than settling to a number, either as $x$ approaches a finite point (an infinite limit, signalling a vertical asymptote) or as $x$ itself grows without bound (a limit at infinity, potentially signalling a horizontal asymptote). In both cases the symbol $\infty$ describes unbounded behaviour, never an actual numeric value the limit equals — such a limit is, strictly, non-existent. For a ratio of polynomials specifically, comparing the degrees of numerator and denominator instantly predicts whether the limit at infinity is infinite, zero, or the ratio of the leading coefficients.

Two further tools handle limits that resist direct algebra entirely: the Sandwich (Squeeze) Theorem, which pins down the limit of a function trapped between two simpler functions sharing a common limit, and a toolbox of standard limits — $\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$, $\lim_{\theta\to0}\frac{1-\cos\theta}{\theta}=0$, and the exponential/logarithmic family built around the number $e$ — that recur throughout the rest of calculus and are simply memorised as known building blocks rather than re-derived every time they are needed. …