Introduction to the epsilon-delta definition of limits
In accordance with many standard textbooks of calculus, we explain the epsilon-N definition of the limits of sequences before the epsilon-delta definition of the limits of functions. This is the best way to understand the essence of the epsilon-delta definition of limits.
Chapter 1 Epsilon-N Definition
First, we explain the underlying basic idea of the epsilon-N definition of the limits of sequences through the parable of an airplane approaching an airport. Next, we introduce the epsilon-N definition of the limits of sequences, which lays down the basis for rigorous studies of the limits of sequences. Finally, we show simple examples in which the epsilon-N definition is used in arguments concerning convergent sequences.
Chapter 2 Epsilon-Delta Definition
We can analogously understand the epsilon-delta definition of the limits of functions if we understand the epsilon-N definition of the limits of sequences. We introduce the epsilon-delta definition of the limits of functions, which lays down the basis for rigorous studies of the limits of functions. Next, we use the epsilon-delta definition to propose a rigorous definition of the continuity of functions.