Homework
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Homework Assignment #3
Read Section 2.4
Page 91, Exercises: 1 – 33 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limits using the Limit Laws and the
following where c and k are constants.
1. lim x
x9
lim x  9
x9
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limits using the Limit Laws and the
following where c and k are constants.
5. lim  3x  4 
x3
lim  3x  4   3  3  4  9  4  5
x3
lim  3x  4   5
x3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limits using the Limit Laws and the
following where c and k are constants.
9. lim  3t  14 
t 4
lim  3t  14   3  4   14  12  14  2
t 4
lim  3t  14   2
t 4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limits using the Limit Laws and the
following where c and k are constants.
13. lim x  x  1 x  2 
x 2
lim x  x  1 x  2   2  2  1 2  2   2  3  4  24
x 2
lim x  x  1 x  2   24
x 2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limits using the Limit Laws and the
following where c and k are constants.
1 x
17. lim
x 3 1  x
1  x 1  3 2
1
lim



x 3 1  x
1 3 4
2
1 x
1
lim

x 3 1  x
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limits using the Limit Laws and the
following where c and k are constants.
21. lim  x 2  9 x 3 
x 3
lim  x  9 x
2
x 3
3
   3
2
 9  3
1
 1  27 1 28
 9  9  
 
 27  3 3 3
28
lim  x  9 x  
x 3
3
2
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
Evaluate the limit assuming: lim f  x   3, lim g  x   1
x4
x4
25. lim f  x  g  x 
x4
lim f  x  g  x    31  3
x4
lim f  x  g  x   3
x4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
sin x
29. Can the Quotient Law be used to evaluate : lim
?
x 0
x
Explain.
sin x
1
 lim sin x  lim
If the limit is rewritten as: lim
x 0
x 0
x 0 x
x
1
using the Product Law, we obtain lim which does not
x 0 x
exist, resulting in a nonexistent limit. Using the Quotient
Law, direct substitution yields an indeterminate form,
resulting in another nonexistent limit.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 82
33. Use the Limit Laws and the result lim x  c to show
xc
n
n
that lim x  c for all whole numbers.
x c
lim x n  c n  lim x lim x  lim x  c c  c
x c
x c
x c
x c
 lim x n  c n
x c
P1 : lim x  c  lim x1  c1
x c
x c
Pk : lim x k  c k
x c
Pk 1 : lim x x  c c  lim x
k
x c
k
x c
k 1
c
k 1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 2: Limits
Section 2.4: Limits and Continuity
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
If the graph of a function may be drawn without lifting the
pencil from the page, the graph and thus the function are
said to be continuous , such as at x = c in Figure 1.
A break in the graph, such as at x = c in Figure 2, is
called a discontinuity.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The discussion on the previous slide leads to the following
definition:
If a continuous function is defined on [a, b], and c is any
point on (a, b) then the following conditions exist:
lim f  x   f  a  , lim f  x   f  b  , and
x a
xb
lim f  x   f  c  for all c on [a, b].
xc
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Three conditions must hold for a function to be continuous:
1. lim f  x  exists
xc
2. f  c  exists
3. lim f  x   f  c 
xc
Figures 3 and 4 are graphs of continuous functions.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Some functions contain discontinuities that can be removed
by redefining the piecewise function, such as defining
f (2) = 5 instead of f (2) = 10 for the function graphed in
Figure 5. Such discontinuities are called removable
discontinuities.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
A jump discontinuity is a discontinuity for which the
first condition of continuity is not met, that is:
lim f  x   lim f  x 
x c 
x c 
If either lim f  x   f  c  or lim f  x   f  c  , the
x c
x c
function is said to be one-sided continuous as defined
below:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
A jump discontinuity may be one-sided continuous as in
Figure 6 (A) or neither right– nor left–continuous as in 6(B).
A function with a jump discontinuity may not be both right–
and left–continuous at the jump discontinuity, as it would
no longer pass the vertical line test, meaning it is no longer
a function.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
An infinite discontinuity is one in which one or both onesided limits are infinite. Since infinity is a concept rather
than an actual number, f (2) is not defined in Figure 8 (A)
or 8 (B). Some texts state that the limit at x = 2 in Figure
8 (A) does not exist, although ours does not so state.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 9 illustrates what some refer to as an oscillating
discontinuity. In this case, neither the right– nor left–
sided limits exist at x = 0, so neither does the limit.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 91
2. Find the points of discontinuity and state whether
f (x) is left- or right-continuous, or neither at these
points. At which point does f (x) have a removable
discontinuity?
y





x






Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Functions “built” of functions known to be continuous
are also continuous in accordance with Theorem 1.
Theorem 1 is proven as follows:
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
As shown in the proof of Theorem 2, all polynomials are
continuous.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 10 shows the graphs of some basic functions that are
continuous on their domains. This is true of most basic
functions.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Theorem 3 formally states the continuity of some basic
functions.
Since inverse functions are reflections of the parent
function about the function y = x, the inverses of
continuous functions are also continuous on their
domains, as stated in Theorem 4.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 91
Use the Laws of Continuity and Theorems 2–3 to
show that the function is continuous.
x 2  cos x
12. f  x  
3  cos x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 91
Determine the points at which the function is
discontinuous and state the type of discontinuity.
x2
20. f  x  
x2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 91
Determine the points at which the function is
discontinuous and state the type of discontinuity.
3
26. g  t   3t 2  9t 3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 91
Determine the points at which the function is
discontinuous and state the type of discontinuity.
34. f  x   x  x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Just as products and quotients of continuous functions
are continuous on their domains, compositions of
continuous functions are also continuous as stated in
Theorem 5.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
When a function f (x) is known to be continuous, then we
may state lim f  x   f  c . This method of evaluating a
xc
limit is sometimes called the substitution method as the
value of c is substituted for x in the function to obtain the
limit. The substitution method can not be used on the
greatest integer function shown in Figure 12 as it is not
everywhere continuous.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Sometimes continuous functions are used to model physical
quantities that on a large-scale may mimic a continuous
function, but on a small scale these quantities are
incremental and thus discontinuous. Figure 13 shows two
such functions.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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Homework Assignment #4
Read Section 2.5
Page 91, Exercises: 1 – 33 (EOO)
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company