Answer Guide 2
Math 408C: Unique Numbers 56975, 56980, and 56985
Tuesday, September 8, 2009
Homework problems
Section 2.2
8.
Examining the given graph, we see that:
(a)
lim
x
!
2
R
(
x
) =
°1
.
(b)
lim
x
!
5
R
(
x
) = +
1
.
(c)
lim
x
!
3
°
R
(
x
) =
°1
.
(d)
lim
x
!
3+
R
(
x
) = +
1
:
12.
Examining the graph below, we see that
lim
x
!
a
f
(
x
)
exists for all
a
except
a
=
°
1
and
a
= +1
.
5
4
3
2
1
1
2
3
4
5
2
2
4
6
8
10
x
y
38.
(a) With
h
(
x
) =
tan
x
°
x
x
3
, we use a calculator to compute that
h
(0
:
005) = 0
:
333 34
h
(1) = 0
:
557 41
;
h
(0
:
5) = 0
:
370 42
;
h
(0
:
1) = 0
:
334 67
;
h
(0
:
05) = 0
:
333 67
;
h
(0
:
01) = 0
:
333 35
;
h
(0
:
005) = 0
:
333 34
:
(b) The brief numerical experiment above suggests that
lim
x
!
0
h
(
x
) = 1
=
3
, which is
correct.
1
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(c) However, if you continue the numerical experiment and ask your calculator to eval
uate
h
(
x
)
for successively smaller values of
x
,
it will eventually give false values.
For
example, my computer claims that
h
(1
:
0
±
10
°
14
) = 0
:
0
, which is wrong.
Numerical experiments are often unreliable when faced with limits of the form °
0
=
0
±.
Later in the course, we will learn techniques to correctly evaluate limits like this.
The point of this exercise is not that computers are unreliable. They aren°t. But we do
need critical thinking skills and some knowledge of calculus to know when we can ± and
can°t ± trust what they tell us.
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 Fall '06
 McAdam
 Math, Calculus, lim, lim T, lim t2

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