Mathematical Analysis

Download A First Course in Real Analysis (2nd Edition) (Undergraduate by Murray H. Protter, Charles B. Morrey Jr. PDF

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By Murray H. Protter, Charles B. Morrey Jr.

Many alterations were made during this moment variation of A First path in genuine Analysis. the main seen is the addition of many difficulties and the inclusion of solutions to many of the odd-numbered routines. The book's clarity has additionally been more suitable by means of the additional explanation of the various proofs, extra explanatory feedback, and clearer notation.

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Extra info for A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics)

Example text

4 again, it follows that lim x _ 2 + g2(X) = lim x _ 2 + f(x). To find the limit of g2(X) as x -. 2l using on each occasion the one-sided version. By applying the theorems on limit of a constant, obvious limit, limit of a sum, and limit of a product, we see that lim 3(x + 2)(x 2 + 4) = 3 ,4,(4 + 4) = 96. x-2+ Similarly, lim 2(x 2 + 2x x..... 2+ + 4) = 2(4 + 4 + 4) = 24. ) Now, using limit of a quotient and the limit of composite functions, we obtain lim g2(X) = x ..... 2+ 96 f%24 = ~24 J4 = 2. Hence, limx _ 2 + f(x) = 2 and so lim x_2f(x) = 2.

4) For this value of () we have If(x) - LI < s whenever 0 < x - a < (). This last statement implies that f(x) -+ L as x -+ a+. 4) that If(x) - LI < e whenever 0 < a - x < (). Hence f(x) -+ L as x -+ a", (b) Now assume both one-sided limits exist. Given any e > 0, there are numbers {)I > 0 and ()2 > 0 such that f(x) is defined for 0 < x - a < {)I ' and o < a - x < ()2 and moreover the inequalities If(x) - LI < e whenever 0< x - a< ()I' If (x ) - LI < s 0

For n an even positive integer, the function g: x ~ ~ is continuous for x on [0, 00). For n an odd positive integer, g is continuous for x on (- 00, 00). PROOF. ; 0 and for any E I~ - 01 < E > 0, it follows that whenever [x - 01 < en. Therefore g is continuous on the right at 0 for every n. If n is odd, then g is also continuous on the left at O. 3. One-Sided Limits It is sometimes easier to determine one-sided limits than two-sided limits. The next theorem and corollary show that one-sided limits can be used as a tool for finding ordinary limits.

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