Download 2.1 - Introduction to Limits - FILLED IN.notebook

2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
Chapter 2: Limits of Functions
The concept of limit of function f is one of the fundamental ideas that distinguishes calculus from algebra and trigonometry. In the development of calculus in the 18th century, the limit concept was treated intuitively as is done in Section 2.1, where we regard the function value f(x) as getting close to some number L as x gets close to another number a. The flaw in using this definition is the world close. A scientist may consider a measurement as being close to an exact value L if it is within 10­6 centimeters of L. A marathon runner is close to the finish line when there are 100 yards left in the race. An astronomer sometimes measures closeness in terms of light­
years. Thus, to avoid ambiguities, it is necessary to formulate a definition of limit that does not contain the word close. We do so in Section 2.2, by stating what is traditionally called the ε ­ δ (epsilon minus delta ) definition of limit of a function . The definition is precise and applicable to every situation we wish to consider. Later in the chapter we discuss properties that enable us to find many limits easily, without applying the ε ­ δ definition. In the last section we use limits to define continuous function, a concept that is employed extensively throughout calculus.
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
Let's explore the graph:
What is the value of the function at x = 2?
5 Sizes Of Numbers
x
f(x)
1.9
1.2033333
1.99
1.32003333
1.999
1.33200033
1.999
1.33320000
1.9999
1.33332000
2
UNDEFINE
D
2.1
2.01
1.4700000
1.3467000
2.001
1.33466700
2.0001
1.333466777
2.00001 1.33334667
2.00000
1.33333467
1
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
Limit of a Function: Theorem 2.1
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
One Sided Limits: Theorem 2.2
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
Theorem 2.3
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
2.1 ­ Introduction to Limits ­ FILLED IN.notebook
January 13, 2015
Attachments
5 Sizes Of Numbers.mp3