Problem 1: if they exist. Find, giving reasoning, the limits of the following function/sequence, Find limx→∞ f(x) where f: (0, ∞) → R, f(x) = (b) Find limn→∞ En for (n)neN, (c) (yn)neN, Yn = 100 + (−1)n. - x0 = 0, x³ + x(3 − x)(x + 2) + ³ (x + 1)² − 1 x² Your justification could either be rigorously mathematical - i.e. using the e-8 def- inition from the notes or using mathematical reasoning. For example, a complete solution for h : [0, ∞) → R, h(x) = „+1, as x→∞ would be either: x+1' |h(x) − 1| : Xn = = (i) Rigorously: Let € > 0. Set y = 1/2. Then take x € [0, ∞) such that x > y. Then we have 1/(x + 1) < 1/(y + 1). So 4(n + 1)(2 − n)(n+3) n³ x x+1 x+1 x + 1 = 1 x + 1 < 1 < = 7+1 7 Therefore lim→∞ h(x) = 1. I 1 (ii) Reasoning: Since h(x) = x+1 = 1, I, as x gets larger and larger, gets closer and closer to zero. Therefore the limit is 1. I
Problem 1: if they exist. Find, giving reasoning, the limits of the following function/sequence, Find limx→∞ f(x) where f: (0, ∞) → R, f(x) = (b) Find limn→∞ En for (n)neN, (c) (yn)neN, Yn = 100 + (−1)n. - x0 = 0, x³ + x(3 − x)(x + 2) + ³ (x + 1)² − 1 x² Your justification could either be rigorously mathematical - i.e. using the e-8 def- inition from the notes or using mathematical reasoning. For example, a complete solution for h : [0, ∞) → R, h(x) = „+1, as x→∞ would be either: x+1' |h(x) − 1| : Xn = = (i) Rigorously: Let € > 0. Set y = 1/2. Then take x € [0, ∞) such that x > y. Then we have 1/(x + 1) < 1/(y + 1). So 4(n + 1)(2 − n)(n+3) n³ x x+1 x+1 x + 1 = 1 x + 1 < 1 < = 7+1 7 Therefore lim→∞ h(x) = 1. I 1 (ii) Reasoning: Since h(x) = x+1 = 1, I, as x gets larger and larger, gets closer and closer to zero. Therefore the limit is 1. I
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please show all the steps. Parts A) and B) only
Transcribed Image Text:Problem 1:
if they exist.
Find, giving reasoning, the limits of the following function/sequence,
Find limx→∞ f(x) where f: (0, ∞) → R,
f(x)
=
(b) Find limn→∞ En for (n)neN,
(c) (yn)neN, Yn = 100+ (−1)”.
-
x0 = 0,
x³ + x(3 − x)(x + 2) + ³ (x + 1)² − 1
x²
Your justification could either be rigorously mathematical - i.e. using the e-8 def-
inition from the notes or using mathematical reasoning. For example, a complete
solution for h : [0, ∞) → R, h(x) = „+1, as x→∞ would be either:
x+1'
|h(x) − 1| :
Xn =
=
(i) Rigorously: Let € > 0. Set y = 1/2. Then take x € [0, ∞) such that x > y. Then
we have 1/(x + 1) < 1/(y + 1). So
4(n + 1)(2 − n)(n+3)
n³
x
x+1
x+1
x + 1
=
1
x + 1
<
1
< =
7+1 7
Therefore lim→∞ h(x) = 1.
I
1
(ii) Reasoning: Since h(x) = x+1 = 1,
I, as x gets larger and larger, gets closer
and closer to zero. Therefore the limit is 1.
I
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