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GO01962-Smith-v1.cls
UAE_Math_Grade_12_Vol_1_SE_718383_ch3
50. (a) Compute lim
15. lim x 3 QC: OSO/OVY 16. lim e x July 4, 2016 13:38 sin x 2 and compare your result to lim sin x .
x→∞ e x x→∞ x 4 x→0 x 2 2 x→0 x
(
17. lim x cos x − sin x 18. lim cot x − 1 ) (b) Compute lim 1 − cos x and compare your result to
x
4
x→0
2
x→0 x sin x x→0 x 1 − cos x
( ) lim .
( x + 1 2 ) 1 2
19. lim − 20. lim tan x + x→0 x
x→0 x sin 2x x→ ∕2 x − ∕2 (c) Use your results from parts (a) and (b) to evaluate
ln x ln x lim sin x 3 and lim 1 − cos x 3
21. lim 3 6 without doing any calcula-
x→∞ x 2 22. lim √ x x→0 x x→0 x
x→∞
tions.
23. lim te −t 24. lim t sin (1∕t) 51. Find functions f such that lim f(x) has the indeterminate
t→∞ t→∞ x→∞
ln (ln t) sin (sin t) form ∞ , but where the limit (a) does not exist; (b) equals 0;
∞
25. lim 26. lim (c) equals 3 and (d) equals −4.
t→1 ln t t→0 sin t
sin (sinh x) ( sin x − sinh x ) 52. Find functions f such that lim f(x) has the indeterminate
27. lim 28. lim x→∞
x→0 sinh (sin x) x→0 cos x − cosh x form ∞−∞, but where the limit (a) does not exist; (b)
√ equals 0 and (c) equals 2.
ln x x
29. lim 30. lim
x→0 + cot x x→0 + ln x In exercises 53 and 54, determine which function “dominates,”
√ where we say that the function f dominates the function g as
2
31. lim( x + 1 − x) 32. lim(ln x − x)
x→∞ x→∞ f(x)
√ x → ∞ if f(x) = g(x) =∞ and either =∞ or
( 1 ) x | x + 1 | x 2 −4 x→∞ x→∞ x→∞ g(x)
33. lim 1 + 34. lim | |
x→∞ x x→∞ x − 2 | | g(x)
|
|
( √ ) √ x→∞ f(x) = 0.
1 x 5 − x − 2
35. lim √ −
x→0 + x x + 1 36. lim √ 10 − x − 3 53. e or x (n = any positive integer)
x→1
x
n
37. lim (1∕x) x 38. lim (cos x) 1∕x
x→0 + x→0 + p
54. ln x or x (for any number p > 0)
( t − 3 ) t ( t − 3 ) t ............................................................
39. lim 40. lim
t→∞ t + 2 t→∞ 2t + 1
............................................................ 55. Basedonyouranswertoexercise53,conjecture lim(e t∕2 − t )
3
x→∞
In exercises 41–44, find all error(s). and prove that your conjecture is correct.
√ x − ln x
41. lim cos x = lim −sin x = lim −cos x =− 1 56. Evaluate lim . In the long run, what fraction of
x→0 x 2 x→0 2x x→0 2 2 x→∞ √ x
√ √
e − 1 e x e x 1 x does x − ln x represent?
x
42. lim = lim = lim =
x→0 x 2 x→0 2x x→0 2 2 ln (x + 2x + 1)
3
57. Evaluate lim . Generalize your result to
2
x 2 x 2 2x 2 x→∞ ln (x + x + 2)
43. lim = lim = lim = lim
x→0 ln x 2 x→0 2 ln x x→0 2∕x x→0 −2∕x 2 ln (p(x))
lim for polynomials p and q such that p(x) > 0
2
= lim(−x ) = 0. x→∞ ln (q(x))
x→0 and q(x) > 0 for x > 0.
sin x cos x −sin x
44. lim = lim = lim = 0. ln (e + x)
3x
x→0 x 2 x→0 2x x→0 2 58. Evaluate lim 2x . Generalize your result to
x→∞ ln (e + 4)
............................................................ ln (e + p(x))
kx
lim
In exercises 45–48, name the method by determining whether x→∞ ln (e + q(x)) for polynomials p and q and positive
cx
l’Ĥ opital’s Rule should be used or not. numbers k and c.
csc x x −3∕2 f(x) f(x )
2
x→0 + x x→0 + ln x x→0 g(x) x→0 g(x )
45. lim √ 46. lim 59. If lim = L, what can be said about lim ? Explain
2
2
2
x − 3x + 1 ln(x ) f(x)
47. lim 48. lim why knowing that lim = L for a ≠ 0, 1 does not tell you
x→∞ tan −1 x x→∞ e x∕3 x→a g(x)
............................................................ f(x )
2
anything about lim 2 .
49. (a) Starting with lim sin 3x , cancel sin to get lim 3x , then x→a g(x )
x→0 sin 2x x→0 2x 2
3
cancel x’stoget . This answer is correct. Is either of 60. Give an example of functions f and g for which lim f(x )
2
the steps used valid? Use linear approximations to ar- x→0 g(x )
2
gue that the first step is likely to give a correct answer. exists, but lim f(x) does not exist. Copyright © McGraw-Hill Education
(b) Evaluate lim sin nx for nonzero constants n and m. x→0 g(x)
x→0 sin mx
248 | Lesson 4-2 | Indeterminate Forms and L’Hôpital’s Rule