Math 1100-006
§5.3 Homework
Solution Key
1. (2 points) (Problem 1) For each given x-value, use the figure to determine whether the
function is continuous or discontinuous at that x-value. If the function is discontinuous,
state which of the three conditions that define continuity is not satisfied.
(a) x = −5
(b) x = 1
(c) x = 3
(d) x = 0
Solution:
(a) The function is continuous at x = −5.
(b) The function is discontinuous at x = 1 because f (1) is undefined.
(c) The function is discontinuous at x = 3 because limx→3 f (x) DNE.
(d) The function is discontinuous at x = 0 because limx→0 f (x) DNE, and f (0) is
undefined.
2. (2 points) (Problem 27)
(a) Use analytic methods to evaluate the limit:
x3 − 1
x→+∞ x3 + 4
lim
(b) What does the result from (a) tell you about horizontal asymptotes?
Solution:
(a)
x3 − 1
= lim
lim 3
x→+∞
x→+∞ x + 4
1/x3
1/x3
x3 − 1
x3 + 4
1 − 1/x3
x→+∞ 1 + 4/x3
1
=
1
=1
= lim
(b) The limit in part (a) tells you that f (x) =
the line y = 1.
x3 −1
x3 +4
has a horizontal asymptote to
3. (2 points) (Problem 31)
(a) Use analytic methods to evaluate the limit:
3x2 + 5x
lim
x→+∞ 6x + 1
(b) What does the result from (a) tell you about horizontal asymptotes?
Solution:
(a)
3x2 + 5x
lim
= lim
x→+∞ 6x + 1
x→+∞
1/x2
1/x2
3x2 + 5x
6x + 1
3 + 5/x
DNE
x→+∞ 6/x + 1/x2
= lim
Since the numerator (3 + 5/x) approaches 3 as x approaches +∞, while the
denominator (6/x + 1/x2 ) approaches 0, the limit is undefined as x approaches
+∞.
(b) Since the limit in part (a) does not exist, the function f (x) =
have any horizontal asymptotes.
3x2 +5x
6x+1
does not
4. (2 points) (Problem 46) Suppose that the cost C of removing p percent of the particulate
pollution from the exhaust gases at an industrial site is given by
C(p) =
8100p
100 − p
Describe any discontinuities for C(p). Explain what each discontinuity means.
Solution: C(p) is a rational function, so is potentially discontinuous when the denominator (100 − p) is zero, which happens when p = 100. This means that it would
be impossible to remove 100 percent of the pollution from the exhaust gases at this
particular site.
5. (2 points) (Problem 47) The precent p of particulate pollution that can be removed from
the smokestacks of an industrial plant by spending C dollars is given by
p=
100C
7300 + C
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Find the percent of the polution that could be removed if spending were allowed to
increase without bound. Can 100% of the pollution be removed? Explain.
Solution: We want to find the limit of the rational function
+∞:
100C
7300+C
as C approaches
1/C
100C
100C
= lim 1
C→+∞ /C 7300 + C
C→+∞ 7300 + C
100
= lim 7300
C→+∞
/C + 1
100
=
1
lim
So, if spending were allowed to increase without bound, any percentage p which is
less than 100 could be removed. (This is because there is a horizontal asymptote to
p = 100, so the value of p gets as close as we want to 100, but doesn’t ever reach 100.
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