Page 71 - Schaum's Outline of Theory and Problems of Advanced Calculus

62 FUNCTIONS, LIMITS, AND CONTINUITY [CHAP. 3

cos x
3.60. Prove that (a) lim 10 x ¼ 0; lim ¼ 0:
ðbÞ
x! 1 x þ
x!1
3.61. Explain why (a) lim sin x does not exist, (b) lim e x sin x does not exist.
x!1 x!1
3.62. If f ðxÞ¼ 3x þjxj ,evaluate (a) lim f ðxÞ; ðbÞ lim f ðxÞ; ðcÞ lim f ðxÞ; ðdÞ lim f ðxÞ;
x!1 x! 1 x!0þ x!0
7x 5jxj
ðeÞ lim f ðxÞ.
x!0
Ans.(a)2, (b)1/6, (c)2, (d)1/6, (e) does not exist.
3.63. If ½xŠ¼ largest integer @ x,evaluate (a) lim fx ½xŠg; ðbÞ lim fx ½xŠg.
Ans.(a)0, (b)1 x!2þ x!2

2
2
3
A.
3.64. If lim f ðxÞ¼ A,prove that (a) lim f f ðxÞg ¼ A , (b) lim p ffiffiffiffiffiffiffiffiffi p 3 ffiffiffiffi
f ðxÞ ¼
x!x 0 x!x 0 x!x 0
What generalizations of these do you suspect are true? Can you prove them?
3.65. If lim f ðxÞ¼ A and lim gðxÞ¼ B,prove that
x!x 0 x!x 0
lim f f ðxÞ gðxÞg ¼ A B; ðbÞ lim faf ðxÞþ bgðxÞg ¼ aA þ bB where a; b ¼ any constants.
ðaÞ
x!x 0 x!x 0
3.66. If the limits of f ðxÞ, gðxÞ; and hðxÞ are A, B; and C respectively, prove that:
(a) lim f f ðxÞþ gðxÞþ hðxÞg ¼ A þ B þ C, (b) lim f ðxÞgðxÞhðxÞ¼ ABC. Generalize these results.
x!x 0 x!x 0
3.67. Evaluate each of the following using the theorems on limits.
( 2 )
2x 1 2 3x
lim Ans: ðaÞ 8=21
ðaÞ 2
x 5x þ 3
x!1=2 ð3x þ 2Þð5x 3Þ
lim ð3x 1Þð2x þ 3Þ ðbÞ 3=10
ðbÞ x!1 ð5x 3Þð4x þ 5Þ
3x 2x

lim ðcÞ 1
x! 1 x 1 x þ 1
ðcÞ
1 1 2x
lim ðdÞ 1=32
x!1 x 1 x þ 3 3x þ 5
ðdÞ
3 8 þ h 2
p ffiffiffiffiffiffiffiffiffiffiffi
3
3.68. Evaluate lim . (Hint: Let 8 þ h ¼ x Þ. Ans. 1/12
h!0 h
A
3.69. If lim f ðxÞ¼ A and lim gðxÞ¼ B 6¼ 0, prove directly that lim f ðxÞ ¼ .
x!x 0 x!x 0 x!x 0 gðxÞ B
sin x
3.70. Given lim ¼ 1, evaluate:
x!0 x
sin 3x 1 cos x 6x sin 2x 1 2cos x þ cos 2x
lim ðcÞ lim ðeÞ lim ðgÞ lim
ðaÞ 2 2
x!0 x x!0 x x!0 2x þ 3 sin 4x x!0 x
1 cos x cos ax cos bx 3 sin x sin 3 x
lim ðdÞ limðx 3Þ csc x ð f Þ lim ðhÞ lim
ðbÞ 2 3
x!0 x x!3 x!0 x x!1 x
2
2
Ans.(a)3, (b)0, (c)1/2, (d) 1= , (e)2/7, ( f ) 1 ðb a Þ, (g) 1, (h)4 3
2
x
e 1
3.71. If lim ¼ 1, prove that:
x!0 x
x
e ax e bx a b x a tanh ax
lim ¼ b a; ðbÞ lim ¼ ln ; a; b > 0; ðcÞ lim ¼ a:
ðaÞ
x!0 x x!0 x b x!0 x