Page 70 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 3] FUNCTIONS, LIMITS, AND CONTINUITY 61

3.46. (a)Prove that tan 1 x þ cot 1 x ¼ =2ifthe conventional principal values on Page 44 are taken. (b)Is
1
tan 1 x þ tan ð1=xÞ¼ =2also? Explain.

x þ y
3.47. If f ðxÞ¼ tan 1 x,prove that f ðxÞþ f ðyÞ¼ f ,discussing the case xy ¼ 1.
1 xy
3.48. Prove that tan 1 a tan 1 b ¼ cot 1 b cot 1 a.
3.49. Prove the identities:
3
2
3
2
1
(a)1 tanh x ¼ sech x, (b) sin 3x ¼ 3 sin x 4 sin x, (c)cos 3x ¼ 4cos x 3cos x, (d)tanh x ¼
2
1
ðsinh xÞ=ð1 þ cosh xÞ, (e)ln jcsc x cot xj¼ ln j tan xj.
2
2
3.50. Find the relative and absolute maxima and minima of: (a) f ðxÞ¼ðsin xÞ=x, f ð0Þ¼ 1; (b) f ðxÞ¼ ðsin xÞ=
2
x , f ð0Þ¼ 1. Discuss the cases when f ð0Þ is undefined or f ð0Þ is defined but 6¼ 1.
LIMITS
3.51. Evaluate the following limits, first by using the definition and then using theorems on limits.
2
4
1 x 4 p x 2 ð2 þ hÞ 16
ffiffiffi
2
limðx 3x þ 2Þ; lim ; ðcÞ lim ; ðdÞ lim ; ðeÞ lim ;
ðaÞ ðbÞ
x!3 x! 1 2x 5 x!2 x 2 x!4 4 x h!0 h
ffiffiffi
p
x
lim :
ð f Þ
x!1 x þ 1
1
1
Ans. ðaÞ 2; ðbÞ ; ðcÞ 4; ðdÞ ; ðeÞ 32; 1
7 4 ð f Þ 2
8
3x 1; x < 0
<
3.52. Let f ðxÞ¼ 0; x ¼ 0 : ðaÞ Construct a graph of f ðxÞ.
2x þ 5; x > 0
:
Evaluate (b) lim f ðxÞ; ðcÞ lim f ðxÞ; ðdÞ lim f ðxÞ; ðeÞ lim f ðxÞ; ð f Þ lim f ðxÞ,justifying your
x!2 x! 3 x!0þ x!0 x!0
answer in each case.
Ans. (b)9, (c) 10, (d)5, (e) 1, ( f ) does not exist
3.53. Evaluate (a) lim f ðhÞ f ð0þÞ and (b) lim f ðhÞ f ð0 Þ , where f ðxÞ is the function of Prob. 3.52.
h h
h!0þ h!0
Ans. (a)2, (b)3
2
3.54. (a)If f ðxÞ¼ x cos 1=x,evaluate lim f ðxÞ,justifying your answer. (b) Does your answer to (a) still remain
x!0
2
the same if we consider f ðxÞ¼ x cos 1=x, x 6¼ 0, f ð0Þ¼ 2? Explain.
2
3.55. Prove that lim 10 1=ðx 3Þ ¼ 0using the definition.
x!3
1 þ 10 1=x
1
3.56. Let f ðxÞ¼ , x 6¼ 0, f ð0Þ¼ . Evaluate (a) lim f ðxÞ, (b) lim f ðxÞ, (c) lim f ðxÞ,justifying
2 10 1=x 2 x!0þ x!0 x!0
answers in all cases.
1
Ans. (a) , (b) 1; ðcÞ does not exist.
2
3.57. Find (a) lim jxj ; ðbÞ lim jxj . Illustrate your answers graphically.
x!0þ x x!0 x
Ans. (a)1, (b) 1
3.58. If f ðxÞ is the function defined in Problem 3.56, does lim f ðjxjÞ exist? Explain.
x!0
3.59. Explain exactly what is meant when one writes:
2 x 2x þ 5 2
lim ¼ 1; lim ð1 e 1=x Þ¼ 1; ðcÞ lim ¼ :
2 x!1 3x 2 3
ðaÞ ðbÞ
x!0þ
x!3 ðx 3Þ