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
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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!ThanksLet 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.Investigate the limit numerically and graphically. lim?→±∞8?+14?2+9‾‾‾‾‾‾‾‾√limx→±∞8x+14x2+9 Calculate the values of ?(?)=8?+14?2+9‾‾‾‾‾‾‾‾√f(x)=8x+14x2+9 for ?=±100,x=±100, ±500,±500, ±1000,±1000, and ±10000.and ±10000. (Use decimal notation. Give your answers to six decimal places.) ?(−100)=f(−100)= ?(−500)=f(−500)= ?(−1000)=f(−1000)= ?(−10000)=f(−10000)= ?(100)=f(100)= ?(500)=f(500)= ?(1000)=f(1000)= ?(10000)=f(10000)= Graph ?(?)f(x) using the graphing utility. ?(?)=f(x)= 8x+1√4x2+9 powered by What are the horizontal asymptotes of ??of f? (Give your answer as a comma‑separated list of equations. Express numbers in exact form. Use symbolic notation and fractions where needed.) horizontal asymptote(s):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.