James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 2.2 The Limit of a Function 93
(f) The equations of the vertical asymptotes.
y
17. lim f sxd − 4, lim f sxd − 2, lim
x l 31 x l 32 f s3d − 3, f s22d − 1
x l 22
f sxd − 2,
_7
_3
0 6
x
18. lim f sxd − 2, lim f sxd − 0, lim f sxd − 3,
x l 02 x l 01 x l 42 lim f sxd − 0, f s0d − 2, f s4d − 1
x l 41 ;
10. A patient receives a 150-mg injection of a drug every
4 hours. The graph shows the amount f std of the drug in
the blood stream after t hours. Find
lim f std and lim f std
tl 122 tl 12 1
and explain the significance of these one-sided limits.
f(t)
300
150
0
4 8 12 16 t
11–12 Sketch the graph of the function and use it to determine
the values of a for which lim x l a f sxd exists.
1 x
11. f sxd −H1 x 2
2 2 x
12. f sxd −H1 1 sin x
cos x
sin x
if x , 21
if 21 < x , 1
if x > 1
if x , 0
if 0 < x <
if x .
13–14 Use the graph of the function f to state the value of
each limit, if it exists. If it does not exist, explain why.
(a) lim
x l 0 2 f sxd
13. f sxd −
(b) lim
x l 0 1 f sxd
(c) lim
x l 0
f sxd
1
1 1 e 14. f sxd − x 2 1 x
1yx
sx 3 1 x 2
15–18 Sketch the graph of an example of a function f that
satisfies all of the given conditions.
15. lim f sxd − 21, lim f sxd − 2, f s0d − 1
x l 02 x l 01 16. lim f sxd − 1, lim f sxd − 22, lim f sxd − 2,
x l 0 x l 32 x l 31 f s0d − 21, f s3d − 1
;
19–22 Guess the value of the limit (if it exists) by evaluating
the function at the given numbers (correct to six decimal places).
x 2 2 3x
19. lim
x l3 x 2 2 9 ,
x − 3.1, 3.05, 3.01, 3.001, 3.0001,
2.9, 2.95, 2.99, 2.999, 2.9999
x 2 2 3x
20. lim
x l23 x 2 2 9 ,
x − 22.5, 22.9, 22.95, 22.99, 22.999, 22.9999,
23.5, 23.1, 23.05, 23.01, 23.001, 23.0001
e 5t 2 1
21. lim , t − 60.5, 60.1, 60.01, 60.001, 60.0001
tl 0 t
s2 1 hd 5 2 32
22. lim
,
hl 0 h
h − 60.5, 60.1, 60.01, 60.001, 60.0001
23–28 Use a table of values to estimate the value of the limit.
If you have a graphing device, use it to confirm your result
graphically.
23. lim
x l 4
ln x 2 ln 4
x 2 4
25. lim
l 0
sin 3
tan 2
27. lim
x l0 1 x x
24. lim
p l 21
1 1 p 9
1 1 p 15
26. lim
t l 0
5 t 2 1
t
28. lim
x l0 1 x 2 ln x
29. (a) By graphing the function f sxd − scos 2x 2 cos xdyx 2
and zooming in toward the point where the graph
crosses the y-axis, estimate the value of lim x l 0 f sxd.
(b) Check your answer in part (a) by evaluating f sxd for
values of x that approach 0.
; 30. (a) Estimate the value of
lim
x l 0
sin x
sin x
by graphing the function f sxd − ssin xdyssin xd.
State your answer correct to two decimal places.
(b) Check your answer in part (a) by evaluating f sxd for
values of x that approach 0.
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