Use the definition to show that x2 + x -6 lim = 5. x - 2 4.
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4
![Use mathematical induction to show that
>(4k – 1) = n(2n + 1)
k=1
for all n > 1.
2. .
Let U = (0, 1) U (2, 3) U (4, 5) be a union of three open
intervals and A= [0, 10] \ U. Show that A is compact.
Use the definition to show that
4n – 5
lim
n+00 2n + 3
= 2.
4.
Use the definition to show that
x + x – 6
lim
= 5.
x - 2
5.
Calculate the following limits
lim n(vn2 +1-n),
lim
0+ sin
Show that the function f(x) = a is uniformly continuous
n00
6.
on [10, 20] but not uniformly continuous on [10, +o0).
(2x, +7)/4
Define a sequence {xn} by x1=2 and xn+1=
for n = 1, 2, 3, .. Show that {rn} is convergent and find its limit.
7.
8.
Show that cos a- cos y < |æ – y| for all a and y.
3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bf7da54-5138-4b57-a0f9-aec30fa8e8cf%2F7c67c98e-0617-4b0d-8796-bfde837ca35a%2Fzrskwlb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Use mathematical induction to show that
>(4k – 1) = n(2n + 1)
k=1
for all n > 1.
2. .
Let U = (0, 1) U (2, 3) U (4, 5) be a union of three open
intervals and A= [0, 10] \ U. Show that A is compact.
Use the definition to show that
4n – 5
lim
n+00 2n + 3
= 2.
4.
Use the definition to show that
x + x – 6
lim
= 5.
x - 2
5.
Calculate the following limits
lim n(vn2 +1-n),
lim
0+ sin
Show that the function f(x) = a is uniformly continuous
n00
6.
on [10, 20] but not uniformly continuous on [10, +o0).
(2x, +7)/4
Define a sequence {xn} by x1=2 and xn+1=
for n = 1, 2, 3, .. Show that {rn} is convergent and find its limit.
7.
8.
Show that cos a- cos y < |æ – y| for all a and y.
3.
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