i need the answer quickly 101. Sets and EventsExample 1.4.1 As an easy exercise, check that you believe thatlim [0, 1 – 1/n] = [0, 1)lim [0, 1- 1/n) = [0, 1)n 00lim [0, 1+1/n] = [0, 1]lim [0, 1 +1/n) = [0, 1].

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Answered: Example 1.4.1 As an easy exercise,…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

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Let n be a positive integer. For which n are the two infinite one-sided limits limx→0±1/xn^n equal?
C. Suppose that (sn) and (tn) are sequences so that sn = tn except for finitely many values of n. Using the definition of limit, explain why if limn → ∞ sn = s, then also limn → ∞ tn = s. Explain this by just using definition of limit!Thanks
Let an = n n + 1 . Find a number M such that: (a) |an − 1| ≤ 0.001 for n ≥ M. (b) |an − 1| ≤ 0.00001 for n ≥ M. Then use the limit definition to prove that lim n→∞ an = 1.
1. i. Prove directly from the definitions that if g : ℝ → [1, ∞) is a function so that limx → c g(x) = L, then limx → c 1/g(x) = 1/L.ii. Prove the same result of the previous part, using Relating Sequences to Functions.
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C. Suppose that (sn) and (tn) are sequences so that sn = tn except for finitely many values of n. Using the definition of limit, explain why if limn → ∞ sn = s, then also limn → ∞ tn = s.
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Example 1.4.1 As an easy exercise, check that you believe that lim [0, 1- 1/n] = [0, 1) n-00 lim [0, 1- 1/n) = [0, 1) n-00 10, 1+ 1/n]= [0, 1] lim [0, 1+1/n) = [0, 1]. lim %3D n00