Q. 70 Prove聽that聽limx鈫�0+1x=鈭�,聽w... [FREE SOLUTION]

Chapter 1: Q. 70 (page 99)

$Prove that \lim_{x 鈫� 0^{+}} \frac{1}{x} = 鈭�, with these steps : (a) What is the M 鈥� 未 statement that must be shown to prove that \lim_{x 鈫� 0^{+}} \frac{1}{x} = 鈭� ? (b) Argue that x 鈭� (0, 0 + 未) if and only if 0 < x < 未 . (c) Argue that \frac{1}{x} 鈭� (M, 鈭�) if and only if x < \frac{1}{M} . You may assume that M > 0 . (d) Given any particular M > 0, what value of 未 would guarantee that if 0 < x < 未, then x < \frac{1}{M} ? Your answer will depend on M . (e) Put the previous four parts together to prove the limit statement .$

Short Answer

Expert verified

$Part (a) The infinite limit, \lim_{x 鈫� 0^{+}} \frac{1}{x} = 鈭� means that for all M > 0, there exists 未 > 0 such that if x 鈭� (0, 0 + 未), then \frac{1}{x} 鈭� (M, 鈭�) . Part (b) x 鈭� (0, 0 + 未) is equivalent to 0 < x < 未 . Part (c) For M > 0, \frac{1}{x} 鈭� (M, 鈭�) means that M < \frac{1}{x} < 鈭�, which means 0 < x < \frac{1}{M} . Part (d) For a given 未 > 0, choose M = \frac{1}{未} Part (e) Given any 未 > 0, let M = \frac{1}{未} If x 鈭� (0, 0 + 未) means that 0 < x < 未, from part (b) . which is same as 0 < x < \frac{1}{M} from our choice in part (d) . From part (c), we get, M < \frac{1}{x} < 鈭� and thus \frac{1}{x} 鈭� (M, 鈭�) .$

Step by step solution

Part (a) Step 1. The M-δ statement

$The infinite limit, \lim_{x 鈫� 0^{+}} \frac{1}{x} = 鈭� means that for all M > 0, there exists 未 > 0 such that if x 鈭� (0, 0 + 未), then \frac{1}{x} 鈭� (M, 鈭�) .$

Part (b) Step 1. Argument regarding δ

$Now, x 鈭� (0, 0 + 未) is equivalent to 0 < x < 未 .$

Part (c) Step 1. Argument regarding M

$For M > 0, \frac{1}{x} 鈭� (M, 鈭�) means that M < \frac{1}{x} < 鈭�, which means 0 < x < \frac{1}{M} .$

Part (d) Step 1. Relationship between M-δ

$For a given 未 > 0, choose M = \frac{1}{未}$

Part (e) Step 1. Proof of the limit statement

$Given any 未 > 0, let M = \frac{1}{未} If x 鈭� (0, 0 + 未) means that 0 < x < 未, from part (b) . which is same as 0 < x < \frac{1}{M} from our choice in part (d) . From part (c), we get, M < \frac{1}{x} < 鈭� and thus \frac{1}{x} 鈭� (M, 鈭�) .$

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Short Answer

Step by step solution

Part (a) Step 1. The M-δ statement

Part (b) Step 1. Argument regarding δ

Part (c) Step 1. Argument regarding M

Part (d) Step 1. Relationship between M-δ

Part (e) Step 1. Proof of the limit statement

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