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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
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88 94 Chapter 1 • Limits and Continuity
Teaching Tip
Most of the limit problems in this section
can be solved using direct substitution.
Consider teaching this section using direct
substitution as the go-to technique. When
substitution does not work, try algebraic
manipulation, a graphical approach, or a
tabular approach. It is not necessary to
teach each of the theorems presented in
this section one at a time in class because
students will not be expected to write a
proof step by step on the AP ® Exam.
AP® Calc Skill Builder
for Example 7
Finding the Limit of a Power
2 2
Investigate lim ( x − 2x+
1)
x→a
Solution
We can rewrite x 2 − 2x + 1 = (x − 1) 2 , which
leads to
lim ( x − 2x+ 1) = lim (( x −1) )
2 2 2 2
x→a x→a
= lim ( x− 1) = ( a−1)
x→a
4 4
{ 2x 2 if x < 1
33. f (x) =
3x 2 at c = 1
53. Slope of a Tangent 0 if Line t < For c
The Heaviside function, u f (x) = 1
− 1 if x > 1
c (t) =
, is a step
2 x2 function − 1: that is used
1 if t ≥ c
x 3 if x < −1
in electrical engineering to(a) model Findathe switch. slope mThe sec ofswitch the secant is off lineif containing t < c, and the it is on
34. f (x) =
x 2 at c =−1
− 1 if x > −1
if t ≥ c.
points P = (2, f (2)) and Q = (2 + h, f (2 + h)).
(b) Use the result from (a) to complete the following table:
x 2 if x ≤ 0
35. f (x) =
at c = 0
2x + 1 if x > 0
EXAMPLE 6 Finding a Limit
⎧
h of the −0.5Heaviside −0.1 −0.001 Function 0.001 0.1 0.5
{
⎨ x 2 if x < 1
m 0sec
if t < 0
Find lim u
36. f (x) = 2 if x = 1 at c = 1
0 (t), where u 0 (t) =
t→0 1 if t ≥ 0
⎩
−3x + 2 if x > 1
(c) Investigate the limit of the slope of the secant line found in (a)
ORIGINS Oliver Heaviside
Solution Since the Heaviside function changes rules at t = 0, we find the one-sided
as h → 0.
(1850--1925) was a self-taught
limits as t approaches 0.
(d) What is the slope of the tangent line to the graph of f at the
Applications mathematician and Extensions
electrical engineer. For t < 0, lim
point P = (2, f (2))?
InHe Problems developed 37–40, a branch sketchofamathematics
u 0(t) = lim 0 = 0 and for t ≥ 0, lim u 0(t) = lim 1 = 1
graph a function with the given t→0 − t→0− t→0
(e) On the same set of axes, graph f and + t→0
the tangent line + to f at
properties. called operational Answers will calculus vary. in which
P = (2, f (2)).
37.
differential
lim f (x)
equations
= 3;
are
lim
solved
f (x)
by
Since the one-sided limits as t approaches 0 are not equal, lim u 0 (t) does not
= 3; lim f (x) = 1;
t→0
converting x→2 them to algebraic x→3− equations. x→3 + exist. ■
Heaviside f (2) = applied 3; f (3) vector = 1calculus to
54. Slope of a Tangent Line For f (x) = x 2 − 1:
NOW WORK Problem 81.
electrical engineering and,
38. lim f (x) = 0; lim perhaps most
f (x) =−2; lim f (x) =−2;
(a) Find the slope m sec of the secant line containing the
significantly, x→−1 he simplified x→2Maxwell's
− x→2 +
points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)).
equations f (−1) toisthe notform defined; used by f (2) electrical =−2
engineers to this day. In 1902 Heaviside
39. lim f (x) = 4; lim f (x) =−1; lim
(b) Use the result from (a) to complete the following table:
claimed
f (x) = 0;
x→1 there is a layer x→0 surrounding
− x→0 +
Earthf (0) from=−1; which radio f (1) signals = 2 bounce,
h −0.1 −0.01 −0.001 −0.0001 0.0001 0.001 0.01 0.1
allowing the signals to travel around the 2 Find the Limit of a Power and the Limit of a Root
40. lim f (x) = 2; lim f (x) = 0; lim f (x) = 1;
m sec
Earth. x→2 Heaviside's claim x→−1 was proved truex→1 Using the Limit of a Product, if lim f (x) exists, then
in 1923.
f (−1)
The
=
layer,
1;
contained
f (2) = 3
in the
x→c
[ ]
(c) Investigate the limit of the slope of the secant line found 2
ionosphere, is named the Heaviside
lim
layer. The function we discuss here is
[ f x→c (x)]2 = lim[ f (x)
in
·
(a)
f (x)]
as h
=
→
lim
0.
f (x) · lim f (x) = lim f (x)
x→c x→c x→c x→c
In Problems 41–50, use either a graph or a table to investigate
one of his minor contributions to
(d) What is the slope of the tangent line to the graph of f at the
each limit.
Repeated use of this property produces
point P =
the
(−1,
next
f (−1))?
corollary.
mathematics and electrical engineering.
|x − 5|
|x − 5|
41. lim
42. lim
43. lim
x→5 + x − 5
x→5 − x − 5
x→ 12
2x
(e) On the same set of axes, graph f and the tangent line to f
COROLLARY −
Limit of a Power at P = (−1, f (−1)).
If lim f (x) exists and if n is a positive integer, then
PAGE
44. lim
x→ 12
2x 45. lim
+
x→ 23
2x 46. lim
−
x→ 23
2x
+
85 55. (a) Investigate lim cos π x→c by using a table and evaluating the
x→0
[ x
function f (x) = cos π ] n
lim [ f
47. lim |x|−x 48. lim |x|−x
x at
x→c (x)]n = lim f (x)
x→c
x→2 + x→2 −
x =− 1
3 3 2 , − 1 4 , − 1 8 , − 1 10 , − 1 12 ,..., 1
12 , 1 10 , 1 8 , 1 4 , 1 2 .
49. lim x−x 50. lim x−x
x→2 + x→2 −
51. Slope of a Tangent Line For f (x) = 3x 2 (b) Investigate lim cos π by using a table and evaluating the
:
x→0 x
EXAMPLE 7 Finding the Limit of a Power
(a) Find the slope of the secant line containing the points (2, 12)
function f (x) = cos π Find:
x at
and (3, 27).
(b) Find the slope of the secant line containing the points (2, 12)
x =−1, − 1 3 , − 1 5 , − 1 7 , − 1 9 ,..., 1 9 , 1 7 , 1 5 , 1 (a) lim x 5 (b) lim(2x − 3) 3 (c) lim x n n a positive 3 , 1. integer
and (x, f (x)), x = 2.
x→2 x→1 x→c
(c) Compare the results from (a) and (b). What do you conclude
(c) Create a table to investigate the slope of the tangent line to the
( ) 5
about the limit? Why do you think this happens? What is
graph of f at 2 using the result from (b).
Solution (a) lim x 5 = lim x = 2 5 = 32
x→2 x→2
your view about using a table to draw a conclusion about
(d) On the same set of axes, graph f , the tangent line to the graph
[
] 3 [
] 3
limits?
of f at the point (2, 12), and the secant line
(b)
from
lim(2x (a).
− 3) 3 = lim (2x − 3) = lim(2x) − lim 3 = (2 − 3) 3 =−1
x→1 x→1 x→1 x→1
(d) Use technology to graph f . Begin with the x-window
52. Slope of a Tangent Line For f (x) = x 3 :
[ ] n
(c) lim x n = lim x = c
n ■
[−2π, 2π] and the y-window [−1, 1]. If you were finding
x→c x→c
(a) Find the slope of the secant line containing the points (2, 8)
lim f (x) using a graph, what would you conclude? Zoom in
x→0
and (3, 27).
on the graph. Describe what you see. (Hint: Be sure your
The result from Example 7(c) is worth remembering since it is used frequently:
(b) Find the slope of the secant line containing the points (2, 8)
calculator is set to the radian mode.)
and (x, f (x)), x = 2.
56. (a) Investigate lim lim cos π by using a table and evaluating the
(c) Create a table to investigate the slope of the tangent line to the
x n x→0
= c n
x2 x→c
graph of f at 2 using the result from (b).
function f (x) = cos π at x =−0.1, −0.01, −0.001,
where c is a number and n is a positive integer.
(d) On the same set of axes, graph f , the tangent line to the graph
x2 of f at the point (2, 8), and the secant line from (a).
−0.0001, 0.0001, 0.001, 0.01, 0.1. NOW WORK Problem 15.
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94
Chapter 1 • Limits and Continuity
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