Answered: Calculate the following limits, or…

Calculate the following limits, or explain why they diverge to ∞, too, or otherwise. You may use any theorems we have proved in class or on homework. (a) lim n³ n∞ (2n+1)(3n+2) (4n+3)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.2: Properties Of Division

Problem 42E

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Question

Math advance

Calculate the following limits, or explain why they diverge to
∞, too, or otherwise. You may use any theorems we have proved in class
or on homework.
(a) lim
n³
n∞ (2n+1)(3n+2) (4n+3)

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18
Use the formal definition of the limit to prove the statement rigorously. lim(3x² + x) = 4 x+1 Consider the following proof. Let e > O be given. We bound | (3x² + x) - 41 using quadratic factoring. |(3x² + x)-4= |3x² +x-4| = |(3x+4)(x-1)| = |x1||3x + 41 Let 6 = min(1, e). Since |x - 1| > e. The proof is not valid; you need to establish | (3x2 + x) - 4|
5x2 + 3 lim х— 1 x2 + бх — 4
mathematically before reading the proof. (It might be helpful to read the discussion that follows Theorem 2.4 or to look at Appendix A.) THEOREM 3.2 THEIS a function that is defined on I except possibly at the point c. Let I be an open interval that contains the point c and suppose a) The function f has limit L at c if and only if for each sequence {Xn) in I\ {c} that converges to c, the sequence (f (x,)} converges to L. b) Suppose that there are sequences (x,} and (y,} in I\ {c} that converge to c such that {ƒ(xn)} converges to L, and (f(yn)} converges to L2. If Li + L2, then the functionf does not have a limit at c. Proof Let P be the statement "the function f has limit L at c" and let Q be the statement "for each sequence {xn} in I\ {c} that converges to c, the sequence (f(xn)} converges to L". A proof that P Q will be left as an exercise. To prove that Q = P, we will prove the contrapositive. Suppose lim f (x) # L. By the negation of the definition, there exists e>0 such that for…
Suppose that following is true. x2 + 6 if x s 0 if 0
Please see attached. Thanks.
A mistake includes but is not limited to a false statement; a statement that is not a logic consequence of its proceeding statement; a statement that is not usefull for or related to the final results. Prove lim,-2 3x2 – - x = 14. Proof for e > 0, we need 3x2 – x – 14| < e. This is |(x + 2)(3x – 7)|< e SO |æ + 2| < |3x – 7| |3x| + 7 Therefore, we just need to pick 8 = $, then when |x + 2| < 8 = , we have |(x + 2)(3x – 7)| < e whch shows the limit=14.