notes/education/math/MATH1210 (calc 1)/Limits.md

Introduction

Every mathematical function can be thought of as a set of ordered pairs, or an input value and an output value.

• Examples include f(x) = x^2 + 2x + 1, and \{(1, 3), (2, 5), (4, 7)\}.

A limit describes how a function behaves near a point, rather than at that point.*

• As an example, given a well behaved function f(x) and the fact that:
  • f(1.9) = 8.41
  • f(1.999) = 8.99401
  • f(2.1) = 9.61
  • f(2.01) = 9.061
  • f(2.0001) = 9.0006 We can note that the smaller the distance of the input value x to 2, the smaller the distance of the output to 9. This is most commonly described in the terms "As x approaches 2, f(x) approaches $9$", or "As x \to 2, f(x) \to 9."

Limits are valuable because they can be used to describe a point on a graph, even if that point is not present.

Standard Notation

The standard notation for a limit is:

\lim_{x \to a} f(x) = L

• As x approaches a, the output of f(x) draws closer to L. In the above notation, x and a are not necessarily equal.
• When plotted, the hole is located at (a, L).

Definitions

Term	Definition
Well behaved function	A function that is continuous, has a single value, and is defined everywhere.