A First Course in Real Analysis by M. H. Protter C. B. Morrey Jr.

By M. H. Protter C. B. Morrey Jr.

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1 f(x) *17. 18. o(sin(l / x)) does not exist. *19. ox loglxl = o. 20. The function f(x) = x cot x is not defined at x = O. Can the domain of f be enlarged to include x = 0 in such a way that the function is continuous on the enlarged domain? 2 Theorems on limits The basic theorems of calculus depend for their proofs on certain standard theorems on limits. These theorems are usually stated without proof in a first course in calculus. In this section we fill the gap by providing proofs of the customary theorems on limits.

Since we don't know in advance whether x is positive or negative, we cannot multiply by x unless we impose additional conditions. We therefore separate the problem into two cases: (i) x is positive, and (ii) x is negative. The desired solution set can be written as the union of the sets S 1 and S2 defined by Sl = {x: ~ <5 and x> 1O}, {x: ~ <5 and x< o}. 6 Similarly, x E S2 ~ 3 > 5x 3 x <"5 ~ x

F(x) 11. 1 12. f(x) 13. 14. *15. 1 16. 1 f(x) *17. 18. o(sin(l / x)) does not exist. *19. ox loglxl = o. 20. The function f(x) = x cot x is not defined at x = O. Can the domain of f be enlarged to include x = 0 in such a way that the function is continuous on the enlarged domain? 2 Theorems on limits The basic theorems of calculus depend for their proofs on certain standard theorems on limits. These theorems are usually stated without proof in a first course in calculus. In this section we fill the gap by providing proofs of the customary theorems on limits.

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