Page 73 - Schaum's Outline of Theory and Problems of Advanced Calculus

64 FUNCTIONS, LIMITS, AND CONTINUITY [CHAP. 3

(b) Illustrate the result in (a)graphically by constructing the graphs of y ¼ tan x and y ¼ x and locating
their points of intersection.
(c)Determine the value of the smallest positive root of tan x ¼ x.
Ans. ðcÞ 4.49 approximately
3.89. Prove that the only real solution of sin x ¼ x is x ¼ 0.
3.90. (a)Prove that cos x cosh x þ 1 ¼ 0 has infinitely many real roots.
(b)Prove that for large values of x the roots approximate those of cos x ¼ 0.
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3.91. Prove that lim x sinð1=xÞ ¼ 0.
x!0 sin x
3.92. Suppose f ðxÞ is continuous at x ¼ x 0 and assume f ðx 0 Þ > 0. Prove that there exists an interval
ðx 0 h; x 0 þ hÞ, where h > 0, in which f ðxÞ > 0. (See Theorem 5, page 47.) [Hint: Show that we can
1
1
make j f ðxÞ f ðx 0 Þj < f ðx 0 Þ. Then show that f ðxÞ A f ðx 0 Þ j f ðxÞ f ðx 0 Þj > f ðx 0 Þ > 0.]
2
2
3.93. (a)Prove Theorem 10, Page 48, for the greatest lower bound m (see Problem 3.34). (b)Prove Theorem 9,
Page 48, and explain its relationship to Theorem 10.