Chapter 1
Limits and Their Properties
Unit Outcomes – At the end of this unit you
will be able to:
•Understand what calculus is and how it
differs from precalculus
•Understand that the tangent line and area
problems are basic to calculus
•Estimate a limit numerically and graphically
•Determine when a limit does not exist
• Learn and use a formal definition of a limit
• Use properties of limits to evaluate limits
• Develop and use a strategy for finding
limits
• Evaluate limits by “dividing out” and
“rationalizing”
• Evaluate a limit using the “Squeeze
Theorem”
• Determine continuity at a point and
continuity on an open interval
• Determine one-sided limits and continuity
on a closed interval
Use properties of continuity
Know and use the Intermediate Value
Theorem
Determine infinite limits from the left and
the right
Find and sketch the vertical asymptotes of
the graph of a function
1.1 A Preview of Calculus
What is
Calculus????????????????????
What Is Calculus
•
•
One answer is to say it is a "limit
machine"
Involves three stages
1. Precalculus/algebra mathematics process
•
Building blocks to produce calculus techniques
2. Limit process
•
The stepping stone to calculus
3. Calculus
•
Derivatives, integrals
Contrasting Algebra & Calculus
• Use f(x) to find the
height of the curve at
x=c
• Find the limit of f(x) as
x approaches c
Contrasting Algebra & Calculus
• Find the average rate
of change between
t = a and t = b
• Find the
instantaneous rate of
change at t = c
Contrasting Algebra & Calculus
• Area of a rectangle
• Area between two
curves
A Preview of Calculus (cont’d)
•Calculus is a branch of mathematics that deals
with rates of change like velocity and
acceleration. What we know of Calculus today,
began in the 17th century with Newton and
Leibnitz
•Calculus deals primarily with limits, derivatives
and integrals
•How does Calculus differ from Precalculus?
Precalculus is static while Calculus is dynamic
See the chart on page 43
On the straight incline the slope remains
the same, therefore the force that must
be used to push it up the hill remains
static. On the curved incline, however,
the slope does not remain the same, so
the force changes and therefore is
dynamic
Examples taken from What is Calculus http://media.wiley.com/product_data/
Two problems are basic to the study of calculus:
The tangent line problem and the area problem
The Tangent Line Problem
The graph of a linear equation has a
constant slope, but the graph of a
quadratic equation does not. So, to find
the slope of the curve at a certain point,
we find the slope of the tangent line at that
point
TANGENT LINE
How will we determine the slope of the tangent line?
Tangent Line Problem
• Approximate slope of tangent to a line
– Start with slope of secant line
We begin by drawing a secant line, then bring the
point of intersection closer and closer to the point
of tangency. This helps us to get a good
approximation of the slope of the tangent line
The Tangent Line Problem
So, as Δx gets smaller the slope gets
smaller and best approximates the slope of
the tangent line
There is a limit to how small the slope can
be.
f  c  x   f  c 
lim
x  0
x
The Area Problem
• In other words!!!! As we increase the
number of rectangles, we get closer and
closer to the actual area of under the
curve!!
• Or we could say “as the limit of the
number of rectangles approaches
infinity”!!!! “we get closer and closer to the
actual area under the curve!!!!
Ways to Evaluate Limits:
• Graphically – show graph and
arrows traveling from each side of
the x-value to find limit
• Numerically – show table values
from both the right and left of the
x-value to discover limit
• Analytically - algebraically
In AP Calculus, we will be approaching
problems in three different ways:
Analytically (using the equation)
Numerically
Graphically
Finding Limits Graphically and Numerically
Finding Limits Graphically
The informal definition of a limit is “what is
happening to y as x gets close to a
certain number”
Notation for a Limit
This is read: The limit of f of x as x
approaches c equals L.
If we are concerned with the limit of f(x) as we
approach some value c from the left hand side,
we write lim f x
x c

 
One-sided Limits—Left-Hand
Limit
This is read--The limit of f
as x approaches c from the
left.
If we are concerned with the limit of f(x) as we
approach some value c from the right hand side,
we write lim f x
x c

 
One-sided Limits—Right-Hand
Limit
This is read: The limit of f
as x approaches c from the
right.
Definition of a Limit
If the righthand and lefthand limits are
equal and
exist, then the
limit exists.
In order for a limit to exist at c
lim f  x  = lim f  x 
x c
x c
and we write: lim f  x   L


x c
Example 2--Graphically
Look at the graph and notice that
y approaches 2 as x approaches
1 from the left. This is also
true from the right.
Therefore the limit exists
and is 2.
Graphically!
What causes this
discontinuity???
ALGEBRA!
Simplify:
x  4 x  12
lim
2
x 2
x  2x
2
When we are computing limits the
question that we are really asking
is what y value is our graph
“intending to take” as we move on
towards x = 2 on our graph.
We are NOT asking what y value
the graph takes at the point in
question!
Example 2--Numerically
Look at the table and notice that
y approaches 2 as x approaches
1 from the left.(slightly
smaller than 1) This
is also true from the right.
(slightly larger than 1)
Therefore the limit
Exists and is 2.
Finding Limits
EXAMPLE
Determine whether the limit exists. If
it does, compute it.

lim x  7
x 4
SOLUTION
3

Let us make a table of values of x approaching 4 and the corresponding values of x3 – 7
as we approach for both from above (from the right) and below (from the left)
x
x3 - 7
x
x3 - 7
4.1
61.921
3.9
52.319
4.01
57.481
3.99
56.521
4.001
57.048
3.999
56.952
4.0001
57.005
3.9999
56.995
As x approaches 4, it appears that x3 – 7 approaches 57. In terms of our
notation,
lim x3  7  57.
x 4


One more try…..
•
•
•
•
•
Turn on the TI-83/84 or 89
Graph y = 2x + 2
Create a table
Study what happens as x approaches 5.
Make sure your tblset is set to:
Independent “ask”. You can then choose
any x value you like and get its y-value
From the Left
x
f(x)
From the Right
x
f(x)
Example. Evaluate the following limit:
The limit is NOT 5!!! Remember from the discussion
after the first example that limits do not care what the
function is actually doing at the point in question.
Limits are only concerned with what is going on
around the point.
Since the only thing
about the function that
we actually changed
was its behavior at
x = 2 this will not
change the limit.
lim g ( x)  4
x2
Finding Limits
EXAMPLE
For the following function g (x), determine whether or not lim g x  exists.
x 3
If so, give the limit.
SOLUTION
lim g x   2.
We can see that as x gets closer and closer to 3, the values of g(x)
get closer and closer to 2. This is true for values of x to both the right and the
left of 3.
x 3
Limit of the Function
• Note: we can approach a limit from
– left … right …both sides
• Function may or may not exist at that point
• At a
– right hand limit, no left
– function not defined
• At b
– left handed limit, no right
– function defined
a
b
Observing a Limit
• Can be observed on a graph.
Observing a Limit
• Can be observed on a graph.
Find each limit, if it exists.
1.
4
lim f ( x )  DNE
x 2
1 for x  2
f (x)  
1 for x  2
2
-5
5
-2
-4
In order for a limit to exist, the two sides of
a graph must match at the given x-value.
2.
lim  2x  1  1
x 1
D.S.
1.2 Limits: A Numerical and
Graphical Approach
• Thus for Example 1:
•
•
•
lim H (x)
•
x1
lim H (x)  4
x3
does not exist
Non Existent Limits
• f(x) grows without bound
Non Existent Limits
Graphical Example 2
What happens as x
approaches zero?
The limit as x approaches zero does not exist.
1
lim  does not exist
x 0 x
Graphical Example 2
What happens as x
approaches zero?
The limit as x approaches zero does not exist.
1
lim  does not exist
x 0 x
From this graph we can see
that as we move in
towards t=0 the function
starts oscillating wildly
and in fact the
oscillations increases in
speed the closer to t=0
that we get. Recall from
our definition of the limit
that in order for a limit to
exist the function must be
settling down in towards
a single value as we get
closer to the point in
question.
This function clearly
does not settle in
towards a single number
and so this limit does
not exist!
Common Types of Behavior
Associated with Nonexistence of a
Limit
SUMMATION
•
Introduction to limits
– The limit of a function is the y value the
graph is getting closer to as x gets closer to
a particular value
– Making a table of values to calculate the
limit – must be done on a calculator
– Sketch a graph to calculate the limit, or use
an already existing graph to calculate the
limit
When limits fail to exist
1. When the right hand and left hand limits
do not agree
2. When there is unbounded behavior
(as we have just seen)
3. When there is oscillating behavior