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).

Continuity

A function is continuous if their graph can be traced with a pencil without lifting the pencil from the page.

Formally, a function f is continuous at a point a if:

• f(a) is defined

• \lim_{x \to a} f(x) exists

• \lim_{x \to a} = f(a)

• A function is continuous on the open interval (a, b) if it is continuous at all points between a and b

• A function is continuous on the closed interval [a, b] if it is continuous at all points between a and b

Elementary Functions

An elementary function is any function that is defined using:

• Polynomial functions
• Rational functions
• Root functions
• Trig functions
• Inverse trig functions
• Exponential functions
• Logarithmic functions
• Operations of:
  • Addition
  • Subtraction
  • Multiplication
  • Division
  • Composition

A piece-wise function is not considered an elementary function

• If f and g are continuous at a point x = a and c is a constant then the following functions are also continuous at x = a
• If g is continuous at a and f is continuous at g(a), then f(g(a)) is continuous at a
• If f is an elementary function and if a is in the domain of f, then f is continuous at a Together, the above theorems tell us that if a is in the domain of an elementary function, then \lim_{x \to a} f(x) = f(a).

Intermediate Value Theorem

Let f be a continuous function on the interval ${}

Definitions

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