Limit of a Sequence in Real Analysis
A sequence has a limit \( L \) when its terms eventually stay arbitrarily close to \( L \): for every tolerance, some position exists past which every term is inside it. After learning this a student can say what a claim of the form 'as \( n \to \infty \) the values approach \( L \)' asserts, distinguish a sequence that converges from one that merely gets smaller in steps, and recognise that a limit is a single exact number defined by the whole tail rather than any one term. They can also use a squeeze between an under-estimating and an over-estimating sequence to pin the common limit.
Approximating the Instantaneous Rate of Change of a Function in Calculus
This concept introduces the limit of an infinite sequence, the idea that a sequence's terms can approach arbitrarily close to a fixed real number, called the limit, as the index n grows toward infini…