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88 82 Chapter 1 • Limits and Continuity
AP® Exam Tip
It is helpful to be familiar with limits of
ax
the form lim sin( ) before taking the
x→0
exam.
x
ax
lim sin( ) = a
x→0
x
Consider working with variations of this
problem as well, such as
ax a
lim sin( ) =
x→0
bx b
These limits can be found using the
Squeeze Theorem, but familiarity will
save time during the exam.
On completion of Section 4.5, the
student will be able to solve problems
of this form using L’Hôpital’s Rule, but
in the meantime, students may prefer
to rely on these two forms rather than
finding these limits using the Squeeze
Theorem.
AP® CaLC skill builder
for example 4
Investigating a Limit Using a Table and
Technology
x
Investigate lim sin(5 ) using a table.
x→0
x
Solution
sin(5 x)
The domain of the function fx ( ) =
x
is {x | x ≠ 0}. So f is defined everywhere
on the open interval containing 0, except
at 0. We create a table by evaluating the
function at values of x near 0, choosing
values that are slightly less than 0 and
slightly greater than 0.
2x 2 if x < 1
33. f (x) =
3x 2 at c = 1
53. Slope
e x of
−
a
1
Tangent Line For
e x −
f (x)
1
Table 2 suggests that lim = 1 and lim = 1.
1 − 1 if x > 1
2 x2 − 1:
x→0
− x
x→0 + x
x 3 if x < −1
e x − 1 (a) Find the slope m sec of the secant line containing the
34. f (x) =
x 2 at c =−1 This suggests lim = 1. points ■
− 1 if x > −1
x→0
P = (2, f (2)) and Q = (2 + h, f (2 + h)).
x
(b) Use the result from (a) to complete the following table:
x 2 if x ≤ 0
NOW WORK Problem 13 and AP® Practice Problem 6.
35. f (x) =
at c = 0
2x + 1 if x > 0
⎧
h −0.5 −0.1 −0.001 0.001 0.1 0.5
⎨ x 2 if x < 1
m sec
36. f (x) = 2 if x = 1 at c = 1 EXAMPLE 4 Investigating a Limit Using a Table and Technology
⎩
−3x + 2 if x > 1
sin x (c) Investigate the limit of the slope of the secant line found in (a)
Investigate lim using a table.
x→0 x
as h → 0.
(d) What is the slope of the tangent line to the graph of f at the
Applications and Extensions
Solution The domain of the function point P = f (x) (2, f = (2))?
sin x is {x | x = 0}. So, f is defined
In Problems 37–40, sketch a graph of a function with the given
x
(e) On the same set of axes, graph f and the tangent line to f at
properties. Answers will vary.
everywhere in an open interval containing 0, except for 0.
P = (2, f (2)).
37. lim f (x) = 3; lim f (x) = 3; lim f (x) = 1;
x→2 x→3− x→3 + We investigate the one-sided limits of sin x as x approaches 0 by using a graphing
f (2) = 3; f (3) = 1
54. Slope of a Tangent
x
Line For f (x) = x 2 − 1:
calculator to set up a table. When making the table, we choose numbers x (in radians)
38. lim f (x) = 0; lim f (x) =−2; lim slightly f (x) =−2; less than 0 and numbers (a) Find slightly the slope greater m sec than of the 0. secant See Figure line containing 10. the
x→−1 x→2− x→2 +
points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)).
Figure f (−1) 10
The table in Figure 10 suggests that lim f (x) = 1 and lim f (x) = 1. This
is not defined; f (2) =−2
x→0 −
x→0 +
39. lim f (x) = 4; lim f (x) =−1; lim
(b) Use the result from (a) to complete the following table:
f (x) = 0; sin x
x→1 x→0− x→0 + suggests that lim = 1. ■
NOTE f (x) = sin x is an even function,
f (0) =−1;
x
x→0 x
f (1) = 2
h −0.1 −0.01 −0.001 −0.0001 0.0001 0.001 0.01 0.1
so the Y 1 values in Figure 10 are
40. lim f (x) = 2; lim f (x) = 0; lim f (x) = 1;
m sec
NOW WORK Problem 15 and AP® Practice Problem 8.
symmetric x→2
about x = 0.
x→−1 x→1
f (−1) = 1; f (2) = 3
(c) Investigate the limit of the slope of the secant line found
in (a) as h → 0.
In Problems 41–50, use either a graph or a table to 3investigate
Investigate a Limit (d) Using What isa the Graph slope of the tangent line to the graph of f at the
each limit.
The graph of a function can alsopoint helpP us = investigate (−1, f (−1))? limits. Figure 11 shows the graphs
|x − 5|
|x − 5|
41. lim
42. lim
43. of three lim
x→5 + x − 5
x→5 − x − 5
different
x→ 12
2x functions (e) f , g, Onand theh. same Observe set of axes, that in graph eachf function, and the tangent as x line getstocloser
f
−
to c, whether from the left or from at P the = (−1, right, f the (−1)). value of the function gets closer to the
number L. This is the key idea of a limit.
PAGE
44. lim
x→ 12
2x 45. lim
+
x→ 23
2x 46. lim
−
x→ 23
2x
+
85 55. (a) Investigate lim cos π by using a table and evaluating the
Notice in Figure 11(b) that the value x→0 of g atxc does not affect the limit. Notice in
Figure 11(c) that h is not defined function at c, but f (x) the= value cos π of
47. lim |x|−x 48. lim |x|−x
x at h gets closer to the number L for
x sufficiently close to c. This suggests
x→2 + x→2 −
x =− 1 that
3 3 2 , − 1 the limit
4 , − 1 of
8 , − 1 a function
10 , − 1 as
12 ,..., 1 x approaches
12 , 1 10 , 1 8 , 1 4 , 1 c does
not depend on the value, if it exists, of the function at c.
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 y
by using a table and evaluating the
:
y
y x→0 x
(c, g(c))
(a) Find the slope of the secant y 5 line f (x) containing the points (2, 12)
y function 5 g(x) f (x) = cos π x at
y 5 h(x)
and (3, 27).
y
y
(b) Find the slope of the secant line containing the points (2, 12)
x =−1, − 1 3 , − 1 y
5 , − 1 7 , − 1 9 ,..., 1 9 , 1 7 , 1 5 , 1 (c, L)
3 , 1.
L
L
L
and (x, f (x)), x = 2.
y
y
(c) Compare the results y from (a) and (b). What do you conclude
(c) Create a table to investigate the slope of the tangent line to the
about the limit? Why do you think this happens? What is
graph of f at 2 using the result from (b).
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
limits?
of f at the point (2, 12), and the secant line from (a).
c
x
(d) Use technology to graph f . Begin with the x-window
52. Slope of a Tangent Line For f (x) = x 3 x c x
x
x c x
x
:
[−2π, 2π] and the y-window [−1, 1]. If you were finding
(a) f (c) 5 L (b) g(c) Þ L (c) h(c) is not defined
(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
DF Figure and 11 (3, 27).
on the graph. Describe what you see. (Hint: Be sure your
(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 cos π by using a table and evaluating the
(c) Create a table to investigate the slope of the tangent line to the
x→0 x2 graph of f at 2 using the result from (b). EXAMPLE 5 Investigatingfunction a Limit f (x) Using = cos a Graph
NEED TO REVIEW? Piecewise-defined
{ π at x =−0.1, −0.01, −0.001,
(d) On the same set of axes, graph f , the tangent line to the graph
x2 functions are discussed in Section P.1,
3x + 1 if x = 2
p. 8. of f at the point (2, 8), and the secant line
Use
from
a graph
(a).
to investigate lim f
x→2 −0.0001, (x) if f (x) 0.0001, = 0.001, 10 0.01, 0.1. if x = 2 .
This table suggests that the value of
sin(5 x)
fx ( ) = gets closer and closer to
x
5 as x gets very close to 0. That is, there is
sin(5 x)
evidence indicating that limx→0
= 5.
x
Teaching Tip
Students should have become familiar with
graphing constant, linear, and basic quadratic,
cubic, radical, rational, and trigonometric functions
before taking this course. If they are not, consider
how to remediate these concepts, because basic
graphing skills are necessary to complete the
exam on time. There are many videos on the Khan
Academy Web site and others that can guide a
student to a stronger understanding of the basic
functions. Consult the Additional Chapter 1
Resources document for suggested URLs.
TRM Section 1.1: Worksheet 2
This worksheet includes 19 problems that have
students determine limits using graphs.
TRM Section 1.1: Worksheet 3
This worksheet contains 3 problems in which the
student is given a piecewise function. They are
asked to find the one- and two-sided limits for
a particular x-value and then verify the limits by
graphing the function.
82
Chapter 1 • Limits and Continuity
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