Introduction to Limits
What is a
A Geometric Example
Look at a polygon inscribed in a circle
As the number of sides of the polygon
increases, the polygon is getting closer to
becoming a circle.
If we refer to the polygon as an n-gon,
Continuity and One-Sided Limits
Intuitive Look at Continuity
A function without
The graph can be
drawn without lifting the pencil
Continuity at a Point
A function can be discontinuous at a point
A hole in the functio
Great Sand Dunes National Monument, Colorado
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
f a h f a
is called the derivative of
f x =lim
f x h f x
The derivative of f
Most of the techniques of calculus require that functions
be continuous. A function is continuous if you can draw it
in one motion without picking up your pencil.
A function is continuous at a point if the limit is the same
as the value of
The derivative as the slope of the
What is a derivative?
the rate of change of a function
the slope of the line tangent to
The tangent line
slope of a secant line
f(a) - f(x)
What is calculus?
What do you learn in a calculus class?
How do algebra and calculus differ?
You will be able to answer all of these
questions after you finish the course.
10.1 Introduction to Limits
One of the basic concepts to the study of
Chapter 2: Limits and Continuity
Section 2.1 The Limit Process (An Intuitive Introduction)
a. The Limit Process
b. Area of a Region Bounded by a Curve
c. The Idea of a Limit
e. Illustration of a Limit
f. Limits on Various Functions