Calculus and Analytic Geometry I
Cloud County Community College
Fall, 2012
Instructor: Timothy L. Warkentin
Chapter 02: Limits and Continuity
02.01
02.02
02.03
02.04
02.05
02.06
Rates of Change and Tangents to Curves
Limit of a Function and Limit Laws
The Precise Definition of a Limit
One-Sided Limits
Continuity
Limits Involving Infinity; Asymptotes of Graphs
Chapter 02 Overview
• The formal definition of the Limit of a Function is one of
the most intellectually satisfying in all of mathematics. It
depends on the idea that a test is needed to determine if
two quantities are equal. This test says that two
quantities will be equal if their difference can be shown
to be less than any amount whatsoever. This concept of
a limit allowed Calculus (derivatives and definite
integrals) to be established on a mathematically rigorous
foundation and resolved Zeno's paradoxes.
• The sections in this chapter will be presented in the
following order: 02.01, 02.03, 02.02, 02.04, 02.05, 02.06.
02.01: Rates of Change and Tangents
to Curves 1
• The Average Rate of Change (ARC) computed as the
average of the changes between points in a discrete
data set and as the slope of the secant line between the
first and last data points. Example data on next slide
• The ARC using the slope of the secant line. Examples 1
&4
• Finding the Instantaneous Rate of Change (IRC) of a
continuous function using the Difference Quotient (DQ).
Examples 2 & 5
DQ 
f  x1  h   f  x1 
h
• The limit of the DQ and the slope of the tangent line.
Example 3
Slope of Tangent Line  lim
h 0
f  x1  h   f  x1 
h
02.01: Rates of Change and Tangents
to Curves 2
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Date
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
Population
1,470,495
1,690,949
1,769,257
1,880,999
1,801,028
1,905,299
2,178,611
2,246,578
2,363,679
2,477,574
2,688,418
Population Change
0
+220,454
+78,308
+111,742
-79,971
+104,271
+273,312
+67,967
+117,101
+113,895
+210,844
02.02: Limit of a Function and Limit
Laws 1
• Limits laws are used to avoid epsilon/delta proofs.
Example 5
1. lim [ f [ x ]  g [ x ]]  lim f [ x ]  lim g [ x ]
x c
x c
x c
2. lim [ f [ x ]  g [ x ]]  lim f [ x ]  lim g [ x ]
x c
x c
f [ x]
3. lim
x c
g[ x]

lim f [ x ]
x c
lim g [ x ]
x c
, lim g [ x ]  0
x c
x c
4. lim [ k  f [ x ]]  k  lim f [ x ]
x c
x c
r
s
5. lim f [ x ]  ( lim f [ x ])
x c
x c
r
s
02.02: Limit of a Function and Limit
Laws 2
• Many limits may be found by substitution.
• Calculus operations are straightforward when dealing
with polynomial functions.
• If P[x] is a polynomial then lim P [ x ]  P [ c ] (theorem 2,
x c
page 69). Example 6
• When division by zero prevents the limit of a function
from being taken by using substitution the form of the
function must be changed. Examples 7 & 9
• Using software to estimate limits. Example 8
02.02: Limit of a Function and Limit
Laws 3
• The Sandwich theorem sometimes allows very difficult
limits (the meat) to be found. The problem is finding the
bread. Examples 10 & 11
• Two important limits and another limit law found using
the Sandwich theorem.
1. lim sin    0
2. lim cos    1
0
0
6. lim f  x   0  lim f  x   0
x c
x c
02.03: The Precise Definition of a Limit
1
• Understanding absolute value notation. Examples 1 & 4
x  a  b  b  x  a  b
 ab  x  ab
 x  (a  b, a  b)
• Multiplication of an inequality by a negative reverses the
inequality.
02.03: The Precise Definition of a Limit
2
• The Precise Definition of a Limit is a marvel of
mathematical engineering.
• Let f [x] be defined on an open interval about x0 except
possibly at x0 itself. Then
lim f [ x ]  L
x  x0
If, for every ε > 0, there exists a δ > 0 such that
0  x  x0    f [ x ]  L  
Example 2
02.04: One-Sided Limits 1
• The limits discussed in the previous sections are actually
two-sided limits where the function must have the same
limiting value on either side of x0. The absolute value
notation in the limit definition is a compact version of


f
[
x
]

L
and
f
[
x
]

L
lim 
 lim 
  lim f [ x ]  L .
x c
x c
 x c

• Examples 1 & 2
• The epsilon/delta definitions given in the text for right
and left hand limits are precisely the two definitions
implied by the absolute value notation in the
epsilon/delta definition of a limit given in section 3.
02.04: One-Sided Limits 2
• Important limits: Examples 4, 5, & 6
1. lim sin 1 / x  is undefined
x 0
2. lim
sin  x 
1
x
x 0
3. If lim   x   0 then lim
x 0
4. lim
x 0
x 0

sin  
 lim
x 0
1
sin  

sin   x 
 x 
1
1
02.05: Continuity 1
• Continuity at a Point Examples 1, 2 & 3
– Interior Point: A function y = f [x] is continuous at an
interior point c of its domain if
lim f [ x ]  f [ c ]
x c
– Endpoint: A function y = f [x] is right/left continuous at
a left/right endpoint a/b of its domain if
lim  f [ x ]  f [ a ] / lim  f [ x ]  f [ b ]
x a
x b
02.05: Continuity 2
• A function is left/right continuous if the function is
continuous at all points when approached from the
left/right. Example 4
• A function is continuous on an interval if and only if (iff) it
is continuous at every point in the interval. Example 5
• The continuity laws (theorem 8, page 95) are analogous
to the limit laws since continuity depends on the
existence of limits. Examples 6 & 7
• The inverse function of any continuous function is
continuous over its domain.
• The composition of continuous functions is continuous.
Example 8
02.05: Continuity 3
• Another limit law dealing with the composition of
functions can be established using continuity. Example 9
7. lim f [ g [ x ]]  f [ lim g [ x ]], if f is continuous at lim g [ x ]
x c
x c
x c
• Types of Discontinuities
– Jump (also called Step)
– Asymptotic (also called Infinite, Essential) (look like
slip faults or volcanos)
– Removable (also called Point, Hole): The domain can
be extended by use of a piecewise definition so that
the function is continuous. This process is called
“Continuous Extension to a Point”. Example 10
02.05: Continuity 4
• The Intermediate Value Theorem for Continuous
Functions Examples 11 & 12
– If f [ x ] is continuous on [ a , b ] and f min  N  f max
then there is at least one number c  [ a , b ] such
that f [ c ]  N .
02.06: Limits Involving Infinity;
Asymptotes of Graphs 1
• The symbol for infinity (∞) does not represent a Real
number. Rather it is meant to convey the idea of an
amount beyond all finite (large or small) bounds. Thus
the limit of a function at either plus or minus infinity is not
like an ordinary limit where the limit occurs at a specific
point in the domain.
• The precise definition of a limit at infinity rests upon the
idea that two quantities are equal when they are less
different by an amount (epsilon > 0) whatsoever. It is
very similar to the definition used to define the sum of
infinite series.
• The limit laws at infinity are the same as those already
presented. Examples 2 & 3
02.06: Limits Involving Infinity;
Asymptotes of Graphs 2
• Limits at infinity are used to provide a definition for
horizontal asymptotes. Examples 4 – 9
* The line y = b is a horizontal asymptote of f[x] if either
lim f [ x ]  b or lim
x 
f [ x ]  b.
x  
• Rational functions may have horizontal and/or oblique
asymptotes. Asymptotes are found using long division to
produce the form
p1[ x ]
 q [ x ]  r [ x ].
p 2[ x ]
The quotient q[x] is the required asymptote. Example 10
02.06: Limits Involving Infinity;
Asymptotes of Graphs 3
• Limits at Infinity vs. Infinite Limits.
• When the value of a function approaches plus or minus
infinity at some domain value the limit is not a normal
limit because the value approached is not a Real
number. Examples 11, 12 & 13
• Infinite limits provide a precise definition for vertical
asymptotes. Examples 15 – 18
* The line x = a is a vertical asymptote of f[x] if either
lim  f [ x ]   or lim  f [ x ]   .
x a
x a
• Dominant terms and the far left/right behavior of
functions. Example 19