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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2x + 5, a = 1, t: = om 2. 01 1. f(x) 3. 5 4. f(x) 5. 6. 7. 8. 1 35 2: Continuity and limits In Problems 9 through 17 the functions are defined in an interval about the given value of a but not at a. Determine a value 8 so that for the given values of Land €, the statement "If(x) - LI < € whenever 0 < Ix - al < 8" is valid. Sketch the graph of the given function. 9. f(x) 10. 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.

2n. Note that in this example the inequality n! > 2n is false for n = 1, 2, 3. This made necessary the strange definition of our set S. Remark. 22. A set S of real numbers has the modified inductive property if (1) S has a smallest number and (2) (x + 1) E S whenever xES. 22 becomes: if S is a set of natural numbers with the modified inductive property, the S contains all the natural numbers greater than the smallest natural number in S. We see that in Example 2, we may choose for S the set of all natural numbers greater than 3.

Similarly, a function may have a negative infinite limit, and it may have both positive and negative one-sided infinite limits. a- 00. 2 have analogs for infinite limits which in most cases are the same as for finite limits. However, there are differences and some of these are exhibited in the following theorems. 16 (Uniqueness of limits). If a function f on 1R1 to 1R1 becomes infinite as x ~ a, then f does not tend to a finite limit as x ~ a. PROOF. Suppose thatf~oo andf~L, a finite number, as x~a.