Limits

This section develops the notion of a limit carefully, moving from an intuitive, table-and-graph based understanding towards a small set of theorems that let limits be computed mechanically rather than guessed at from numerical evidence.

The section begins (9.2.1) by building intuition through concrete illustrations — watching the output of a function as the input is pushed closer and closer to a target value from both sides — and then states a formal description of what it means for a limit to exist. It separates out the idea of approaching a point from only one side (9.2.2), which becomes essential for functions defined differently on either side of a point, or defined only on one side of it. Once the intuition is secure, a battery of theorems (9.2.3) is introduced that let a limit be evaluated by direct substitution, algebraic manipulation, or by combining known limits — for polynomials, sums, differences, products, quotients and powers. The section then extends the idea to two situations a plain "substitute and see" approach cannot handle: functions whose values grow without bound near a point (infinite limits, 9.2.4), and functions examined as the input itself grows without bound (limits at infinity, 9.2.5) — the latter giving a quick rule of thumb for rational functions (9.2.6) and several real-world applications (9.2.7). Finally, two of the most useful tools in the whole chapter are developed: the Sandwich (Squeeze) Theorem (9.2.8), which evaluates a limit by trapping a function between two simpler functions sharing a common limit, and a set of standard trigonometric, exponential and logarithmic limits (9.2.9, 9.2.10) that recur constantly in later work.