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Then there is an open set H in H that contains and, along with , an interval Œ ; C  for some positive . Since S has a finite covering fH1 ; : : : ; Hn g of sets from H , it follows that S C has the finite covering fH1 ; : : : ; Hn ; H g. This contradicts the definition of . Now we know that D ˇ, which is in S . Therefore, there is an open set Hˇ in H that contains ˇ and along with ˇ, an interval of the form Œˇ ; ˇ C , for some positive . Since Sˇ is covered by a finite collection of sets fH1 ; : : : ; Hk g, Sˇ is covered by the finite collection fH1 ; : : : ; Hk ; Hˇ g.

1 ” (“limx! 1 ”). How would their proofs have to be changed? (d) e 16. 14, show that à  2jxj 1 (a) lim 1 D1 (b) lim D 2 x! 1 x! 1 1 C x x2 (c) lim sin x does not exist x! 1 18. Find limx! 15. 14. 19. x0 20. 1 Functions and Limits 21. x0 22. x0 23. 1 24. 51 x! 1 x! 1 x! 1 25. x/ > a if x > c. x/ D 1. x// D L. 26. x0 C 27. x/: x! x/ both exist in the extended reals and are equal, in which case all three are equal. 30 consider only the case where at least one of L1 and L2 is ˙1. 28. x0 52 Chapter 2 Differential Calculus of Functions of One Variable 29.

X2 / In either case, f is on I . 19), f is increasing on I . 20), f is decreasing on I . In either of these two cases, f is strictly monotonic on I . 4), and f is nonincreasing on I D Œ0; 2. x/ D x 3 is decreasing on . 6). 19. aC/ D ˛. x0 / < M . x/ < M if a < x < x0 . aC/ D 1. If ˛ > 1, let M D ˛ C , where > 0. 21) If a D 1, this implies that f . 1/ D ˛. x/ ˛j < if a < x < a C ı; a. aC/ D ˛. b / D ˇ. x0 / > M . x/ > M if x0 < x < b. b / D 1. If ˇ < 1, let M D ˇ , where > 0. 1/ D ˇ. x/ ˇj < if b ı < x < b; x0 .

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