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

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.

**Read or Download A First Course in Real Analysis (2nd Edition) (Undergraduate Texts in Mathematics) PDF**

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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.