Page 4 - Calc for the Clueless

Chapter 1
The Beginning— Limits

Informal Definition

We will begin at the beginning. Calculus starts with the concept of limits. We will examine this first intuitively
before we tackle the more difficult theoretical definition.

Let us examine

read, ''The limit of f(x) as x goes to a is L."

This means that the closer x gets to a, the closer f(x) gets to L. We will leave the word "close" unspecified until
later.

Example 1—

We will take points near x = 3, smaller than 3, getting closer to 3. We make a small chart showing this.

x 2x

2.5 5

2.9 5.8

2.99 5.98

2.999 5.998

We see that as x approaches 3 from points less than 3, f(x) approaches 6. Notation:

read, "the limit of f(x) as x goes to 3 from the negative side of 3 (numbers less than 3) is 6." We call this the
limit from the left.

If we do the same thing for numbers greater than 3, the chart would look like this:

x 2x

3.2 6.4

3.1 6.2

3.01 6.02

3.001 6.002