Using only the definition of limit (so, without using Arithmetic of Limits), show that limn→∞ 2/n + 3/(n+1) = 0
Using only the definition of limit (so, without using Arithmetic of Limits), show that limn→∞ 2/n + 3/(n+1) = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
B. Using only the definition of limit (so, without using Arithmetic of Limits), show that
limn→∞ 2/n + 3/(n+1) = 0
![Example: Show directly from definition that lim Y/₁² = 0₁
510
Sulution 1: Proceed exactly similarly to
Solution 2: Notice that for nz1,
Thus,
With more
and
1²³/n²-0] = 1¹/₁² <½/n,
50 we
take
N= Y/₂
After some algebra,
and
complicated sequences, this technique of "replacing"
(bounding) w/ something simpler
be quite useful.
Examples Show directly from definition
We want to find conditions
Solution:
2n²+ 34 -21<ε.
n²-2
sume algebra, (*)
can
So In ²-21 = 23/1²=
n
Multiplying together,
We consider the top + buttom separately!
13n+41 ≤ 13n+4n1=17n1
In²-21 = | 11/²2/1
Examples show that the sequence
Solution: Suppose that
Then
Yn² ≤ ½n,
lim
|3n++ | ≤ 7n. 3/2 = 210
22
n²2
So N=21/₂
336
Cun
Remarks This last example will become
that we
will prove
(We showed earlier that
316
VETO, IN sit.
that
so that
becomes | 3n++ | <E.
Sn=S₁
when nz3
(as then
when n23.
lim
316
(as in the Y/n example).
when nz3.
will suffice, by "sulving" as in previous examples.
lim
2n²+ 3n
n²-2
316
when n21
Sn=(-+)" does not converge.
Sn # 1,)
2 = 21² )
37
=2.
much easier with theorems
bit later
(*)
[n>N] => [1(-1)" - 5) <ε].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbae42449-be42-4869-b2b7-300d7dd59342%2Ff3bf5b71-e256-4f35-b861-3dc6edaffed3%2F496pj5v_processed.png&w=3840&q=75)
Transcribed Image Text:Example: Show directly from definition that lim Y/₁² = 0₁
510
Sulution 1: Proceed exactly similarly to
Solution 2: Notice that for nz1,
Thus,
With more
and
1²³/n²-0] = 1¹/₁² <½/n,
50 we
take
N= Y/₂
After some algebra,
and
complicated sequences, this technique of "replacing"
(bounding) w/ something simpler
be quite useful.
Examples Show directly from definition
We want to find conditions
Solution:
2n²+ 34 -21<ε.
n²-2
sume algebra, (*)
can
So In ²-21 = 23/1²=
n
Multiplying together,
We consider the top + buttom separately!
13n+41 ≤ 13n+4n1=17n1
In²-21 = | 11/²2/1
Examples show that the sequence
Solution: Suppose that
Then
Yn² ≤ ½n,
lim
|3n++ | ≤ 7n. 3/2 = 210
22
n²2
So N=21/₂
336
Cun
Remarks This last example will become
that we
will prove
(We showed earlier that
316
VETO, IN sit.
that
so that
becomes | 3n++ | <E.
Sn=S₁
when nz3
(as then
when n23.
lim
316
(as in the Y/n example).
when nz3.
will suffice, by "sulving" as in previous examples.
lim
2n²+ 3n
n²-2
316
when n21
Sn=(-+)" does not converge.
Sn # 1,)
2 = 21² )
37
=2.
much easier with theorems
bit later
(*)
[n>N] => [1(-1)" - 5) <ε].
Expert Solution
Step 1
Given:
We have to show the given limit using only the definition of the limit.
Step by step
Solved in 2 steps
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