Limits of Functions

Mathematical analysis is born from an apparently simple question: what does a function “tend” to when its variable approaches a certain value? The concept of the limit is the rigorous answer to this question and constitutes the foundation on which derivatives, integrals and all the tools of analysis are built. The chapter starts from Zeno’s paradoxes (where infinitely many terms sum to a finite value), introduces the preliminary tools — neighbourhood and accumulation point — which tell us when it makes sense to take a limit, then the intuitive concept of a limit and its four fundamental cases. There follows the classification into determinate and indeterminate forms and the four strategies for resolving the latter: equivalent functions, hierarchy of infinities, factorisation and rationalisation. The chapter closes with the trick $A^B\to e^{B\ln A}$ for exponential forms and the study of asymptotes.

Sections

• Historical opening: Zeno’s paradoxes
• Domain and accumulation points
• The intuitive concept of a limit
• Determinate and indeterminate forms
• Hierarchy of infinities
• Standard limits and equivalent functions
• Factorisation and rationalisation
• Change of variable
• ln A}$
• Asymptotes

Further Study

• From the paradoxes to the rigorous limit

Exercises

• All the worked exercises of the chapter