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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan SullivanAP
88 84 Chapter 1 • Limits and Continuity
Alternate Example
Investigating a Limit
x
Investigate lim | − 1| .
x→1
x −1
Solution
We can investigate a limit such as this one
in three ways: graphically, algebraically, and
numerically.
Graphically
The graph suggests that lim fx ( ) does not
x→1
exist because lim fx ( ) ≠ lim fx ( ).
24
Algebraically
− +
x→1 x→1
22
y
4
2
22
24
| x −1|
fx ( ) = is equivalent to the
x −1
piecewise-defined function.
⎧ −( x −1)
=−1
x < 1
⎪
fx ( ) =
x −1
⎨
⎪ ( x −1)
− = 1 x > 1
⎩
⎪ x 1
lim fx ( ) does not exist because
x→1
lim fx ( ) ≠ lim fx ( ).
− +
x→1 x→1
Using a Table
x f(x)
0.9 −1
0.99 −1
0.999 −1
1.001 1
1.01 1
1.1 1
The table suggests lim fx ( ) does not exist
x→1
because lim fx ( ) ≠ lim fx ( ).
− +
x→1 x→1
2
4
x
2x 2 if x < 1
33. f (x) =
3x 2 at c = 1 THEOREM 53. Slope of a Tangent Line For f (x) = 1
− 1 if x > 1
2 x2 − 1:
The limit L of a function y = f (x) as x approaches a number c exists if and only if
x 3 (a) Find the slope m
if x < −1
sec of the secant line containing the
both one-sided limits exist at c and both one-sided limits are equal. That is,
34. f (x) =
x 2 at c =−1
points P = (2, f (2)) and Q = (2 + h, f (2 + h)).
− 1 if x > −1
lim f (x) = L(b) ifUse andthe only result if from lim(a) f to (x) complete = limthe following x 2 x→c table:
x→c
if x ≤ 0
x→c +
35. f (x) =
at c = 0
2x + 1 if x > 0
NOW
⎧
h WORK −0.5Problems −0.1 25, −0.001 31, and AP® 0.001 Practice 0.1 Problem 0.5 5.
⎨ x 2 if x < 1
m sec
36. f (x) = 2 if x = 1 at c = 1 A one-sided limit is used to describe the behavior of functions such as
⎩
−3x + 2 if x > 1
f (x) = √ √ x − 1 near x = (c) 1. Since Investigate the √domain the limit of of ftheis slope {x|xof ≥the 1}, secant the left-hand line foundlimit,
in (a)
lim x − 1 makes no sense. But as h lim → 0. x − 1 = 0 suggests how f behaves near and
x→1 − x→1 +
Applications y and Extensions
to the right of 1. See Figure(d) 14 and What Table is the5. slope They of suggest the tangent limlinef to(x) the= graph 0. of f at the
point P = (2, f (2))? x→1 +
In Problems 2 37–40, sketch a graph of a function with the given
(e) On the same set of axes, graph f and the tangent line to f at
properties. Answers will vary.
TABLE 5
1
P = (2, f (2)).
37. lim f (x) = 3; lim f (x) = 3; lim f (x) = 1;
x approaches 1 from the right
x→2 x→3− x→3 +
−−−−−−−−−−−−−−−−−−−−−→
f (2) = 3; f (3) = 1
54. Slope of a Tangent Line For f (x) = x 2 − 1:
1 2 3 4 5 6 x
x 1.009 1.0009 1.00009 1.000009 1.0000009 → 1
38. lim f (x) = 0; lim f (x) =−2; lim
=−2;
(a) Find the slope m sec of the secant line containing the
Figure x→−1 14 f (x) = √ f (x) = √ x − 1 0.0949 0.03 0.00949 0.003 0.000949 f (x) approaches 0
xx→2 − 1− x→2 +
points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)).
f (−1) is not defined; f (2) =−2
39. lim f (x) = 4;
x→1
lim f (x) =−1;
x→0− lim f (x) = 0;
x→0 +
(b) Use the result from (a) NOW to complete WORK the AP® following Practicetable:
Problem 1.
f (0) =−1; f (1) = 2
Using numeric tables and/orhgraphs −0.1gives −0.01us−0.001 an idea−0.0001 of what a0.0001 limit might 0.001 0.01 be. That 0.1
40. lim f (x) = 2;
x→2
lim f (x) = 0;
x→−1 lim is, these methods suggest a limit,
f (x) = 1;
m sec
but there are dangers in using these methods, as the
x→1 following example illustrates.
f (−1) = 1; f (2) = 3
(c) Investigate the limit of the slope of the secant line found
EXAMPLE 7 Investigatingin a(a) Limit as h → 0.
In Problems 41–50, use either a graph or a table to investigate
(d) What is the slope of the tangent line to the graph of f at the
each limit.
Investigate lim sin π
x→0 x . 2 point P = (−1, f (−1))?
|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
Solution The −
domain of the function at P = (−1, f (x) f = (−1)). sin π is {x|x = 0}.
x
2
PAGE
44. lim
x→ 12
2x 45. lim
+
x→ 23
2x 46. Suppose lim
−
x→ 23
2x we let
+
85 x approach 55. (a) Investigate zero in thelimfollowing cos π byway:
using a table and evaluating the
x→0 x
function f (x) = cos π TABLE 47. lim6
|x|−x 48. lim |x|−x
x at
x→2 + x→2 −
x approaches 0 from the left
x =− 1
3 3 2 , − 1 4 , − 1 8 , − 1 x
10 , approaches − 1 12 ,..., 1
0 from
12 , 1 10 , the 1 8 , 1 right
4 , 1 2 .
49. lim x−x 50. lim
−−−−−−−−−−−−−−−−−−−−→
x−x
←−−−−−−−−−−−−−−−−−−−−−
x→2 + x→2
x − 1
− 1 − 1 − 1
51. Slope of a Tangent Line For f (x) = 3x 2 (b) Investigate lim cos π 1 1 1 1
→ 0 ←
10 100
by using a table and evaluating the
: 1000 10,000
x→0 x10,000
1000 100 10
f (x)(a) = Find sin π the slope of0the secant line 0 containing0the points (2, 12) 0 f (x) approaches function f (x) 0 = cos π x at 0 0 0 0
x 2
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 Table 6 suggests that lim sin π 3 , 1.
and (x, f (x)), x = 2.
x→0 x = 0. 2
(c) Compare the results from (a) and (b). What do you conclude
(c) Create a table to investigate the slope of the tangent
Now suppose
line to the
we let x approach zero as follows:
about the limit? Why do you think this happens? What is
graph of f at 2 using the result from (b).
TABLE your view about using a table to draw a conclusion about
(d) 7On the same set of axes, graph f , the tangent line to the graph
limits?
of f at the point (2,
x
12),
approaches
and the secant
0 from
line
the
from
left
(a).
x approaches 0 from the right
−−−−−−−−−−−−−−−−−−−−→
(d) Use technology to graph f . Begin with the x-window
52. Slope of a Tangent Line For f (x) = x 3 ←−−−−−−−−−−−−−−−−−−−−−
:
[−2π, 2π] and the y-window [−1, 1]. If you were finding
x − 2 − 2 − 2 − 2 − 2 2 2 2 2 2
→ 0
(a) Find the slope of the secant line containing the points (2, 8)
lim
←
3 5 7 9 11
f (x) using a11
graph, what 9 would you 7 conclude? 5 Zoom3in
x→0
f (x) = sin and (3, 27). 0.707 0.707 0.707 0.707 0.707 f (x) approaches on the 0.707 graph. Describe 0.707 what 0.707 you see. 0.707 (Hint: Be 0.707 sure your 0.707
(b) Findx 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 Table 7 suggests that lim sin π √
2
graph of f at 2 using the result from (b).
function f (x) = cos π x→0 x = 2 2 ≈ 0.707. at x =−0.1, −0.01, −0.001,
(d) On the same set of axes, graph f , the tangent line to the graph
x2 In fact, by carefully selecting x, we can make f appear to approach any number in
of f at the point (2, 8), and the secant line from (a).
−0.0001, 0.0001, 0.001, 0.01, 0.1.
the interval [−1, 1].
Teaching Tip
Have students create their own piecewise
functions on graph paper. Ask them to write 3 or
4 limit questions based upon their (well-labeled)
graph. Point out to students that being able to
write out and explain piecewise functions so
that another student can follow means that they
have to be very clear. Reinforce good habits and
good explanations. Then they can trade with one
another, find the limits, and discuss the solutions
with each other. Khan Academy offers explanatory
videos, as well as a practice video, on piecewise
functions.
WEB SITE
Graphfree: The Graphfree site is an easy tool
for making piecewise graphs to create additional
examples. Make sure you change the plot type to
Piecewise. A link to this resource is available on
the Additional Chapter 1 Resources document,
available for download.
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Chapter 1 • Limits and Continuity
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