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A.3 Commonly Occurring Limits
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A.3
Commonly Occurring Limits
This appendix verifies limits (4)(6) in Theorem 5 of Section 11.1. Limit 4: If x 6 1, lim x n = 0 We need to show that to each P 7 0 there corresn: ponds an integer N so large that x n 6 P for all n greater than N. Since P1>n : 1, while
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100AWL/Thomas_APPp001-034
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Appendices
x 6 1, there exists an integer N for which P1>N 7 x . In other words, xN = x N 6 P. This is the integer we seek because, if x 6 1, then x n 6 x N for all n 7 N . Combining (1) and (2) produces x n 6 P for all n 7 N , concluding the proof. xn Limit 5: For any number x, lim A 1 + n B = e x n:
(2) (1)
Let
n
x an = a1 + n b . Then x x ln an = ln a1 + n b = n ln a1 + n b : x , as we can see by the following application of lHpitals Rule, in which we differentiate with respect to n: lns 1 + x> n d x lim n ln a1 + n b = lim 1> n n: q n: q a
n
= lim
n: q
Apply Theorem 4, Section 11.1, with s x d = e x to conclude that x a1 + b n = an = e ln an : e x . xn = 0 Since n! x n x n xn , n! n! n!
n
x 1 b # a- 2 b 1 + x> n n x = lim = x. n : q 1 + x> n - 1> n2
Limit 6: For any number x, lim
n:
all we need to show is that x n> n! : 0. We can then apply the Sandwich Theorem for Sequences (Section 11.1, Theorem 2) to conclude that x n> n! : 0. The first step in showing that x n> n! : 0 is to choose an integer M 7 x , so that s x > M d 6 1. By Limit 4, just proved, we then have s x > M dn : 0. We then restrict our attention to values of n 7 M . For these values of n, we can write x n x n = # ## # n! 12 M s M + 1 ds M + 2 d # # n
('''''')''''''* s n - M d factors
n x n x nM M MM x = = a b. M! M M!M n M!M n - M
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley
4100AWL/Thomas_APPp001-034
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A.3 Commonly Occurring Limits
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Thus, 0
n x n MM x a b. n! M! M
Now, the constant M M> M! does not change as n increases. Thus the Sandwich Theorem tells us that x n> n! : 0 because s x > M dn : 0.
Copyright 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley