UNI_FOUNDATION · Calculus · ANALYSIS.LIMIT.RIGOR

Limits and continuity, rigorous foundation

Intuition

Understand first, compress later.

Calculus studies change and accumulation through limiting processes.

Switch representations

A limit at a removable discontinuityf(x) = (x² − 4)/(x − 2)

graph of fopen circle: the value 4 is not attainedfilled dot: f(2) = 1approach from left and right

From the left and from the right the values head for the same number: lim f(x) = 4 as x → 2. The open circle says that value is never attained; the filled dot shows what f actually returns at 2, namely 1. A limit describes the approach, not the value — which is why it exists here even though f jumps at that point.

Formal view

You can limits and continuity, rigorous foundation when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.