01
L’Hopital practice - converting indeterminate form
By imitating the technique of the L’Hopital’s Rule example, find the limit of the sequence:
Solution
01
(1) Indeterminate form:
(2) L’Hopital:
Convert:
Change to
and apply L’Hopital:
(3) Take limit:
Therefore
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02
Squeeze theorem
Determine whether the sequence converges, and if it does, find its limit:
(a)
(b) (Hint for (b): Verify that
.)
Solution
06
(a)
(1) Set up squeeze relations:
(2) Apply theorem:
We have:
Therefore:
We conclude that
converges.
(b)
(1) Generate squeeze inequalities:
Observe:
Rewrite RHS:
Raise all terms to
:
(2) Apply squeeze theorem:
Therefore:
Conclude that:
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03
Computing the terms of a sequence
Calculate the first four terms of each sequence from the given general term, starting at
: (a)
(b) (c) (d) (e) (f)
Solution
04
(a)
(b)
(c)
(d)
(e)
(f)
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04
General term of a sequence
Find a formula for the general term (the
term) of each sequence: (a)
(b) (c)
Solution
05
(a)
(b)
(c)
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05
Limits and convergence
For each sequence, either write the limit value (if it converges), or write ‘diverges’.
(a)
(b) (c) (d)
(e)
Solution
02
(a)
(b) diverges (c) (d)
Observe that
as , but for each , the value is below , in the domain of , which is continuous for . (e)
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06
Limits and convergence
For each sequence, either write the limit value (if it converges), or write ‘diverges’.
(a)
(b) (c) (d) (e)
(f) (g) (h)
Solution
03
(a) diverges
(b) (c) diverges (d) (e)
(f) diverges:
(g)
(h)
Use L’Hopital’s Rule:
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07
Limits and convergence
For each sequence, either write the limit value (if it converges), or write ‘diverges’.
(a)
(b) (c) (d) (e)
(f) (g)
Solution
04
(a)
(b) diverges:
(Note that
.)
(c)
: L’Hopital’s Rule:
(d)
by (c), the sign doesn’t affect convergence to
(e)
: Multiply above and below by
:
(f)
: This is a well-known formula for
. If that formula is not used as the definition of , then it would not be circular reasoning to argue as follows:
(g) diverges:
So:
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