Limits and Continuous Functions

Summary:

This Limits and Continuous Functions note discusses limits and continuous functions. It begins with an informal definition of limits and then introduces the formal definition. The author explains that while it is easy to define concepts like square roots precisely, defining the limit of a function requires more detailed explanations.

The definition of a limit is presented as follows: if f(x) is a function and lim f(x) = L as x approaches a, it means that as x gets closer to a (but not equal to a), f(x) gets closer to the value L. This definition is further illustrated with examples demonstrating how to calculate limits by substituting numbers and exploring the precision of measurements.

The chapter then introduces variations on limits, such as left and right limits, which deal with approaching a value from one side only. It also discusses limits at infinity, where x becomes larger and larger, and the behaviour of the function is examined.

Overall, this chapter provides a foundational understanding of limits and their application in analyzing functions. It emphasizes the importance of precise definitions and explores different scenarios to deepen the reader’s intuition about limits and continuous functions.

Excerpt:

Limits and Continuous Functions

1. Informal definition of limits
While it is easy to define precisely in a few words what a square root is (√a is the positive number whose
square is a), the definition of the limit of a function runs over several terse lines. Most people don’t find it very enlightening when they first see it. (See 2.) So we postpone this for a while and fine-tune our intuition for another page.

1.1. Definition of limit (1st attempt). If f is some function, then
lim f(x) = L
x→a
is read as “the limit of f (x) as x approaches a is L.” It means that if you choose values of x that are close but
not equal to a, then f (x) will be close to the value L; moreover, f (x) gets closer and closer to L as x gets
closer and closer to a.