The Student Room Group

Limits of Sequences

Give an example of a pair of sequences a_n and b_n where b is not zero for all n in the natural numbers, such that (as n tends to infinity) lim a_n = 0 and lim b_n =0 and lim(a_n/b_n) = -3

I know of the quotient rule and how
but I am unsure on what I do here where the limit of g(x) is zero?
(edited 9 years ago)
Original post by _JC95
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No need to overcomplicate the question. Just think of sequences that would satisfy the requirements stated in the question.

Hope that helps.
Reply 2
Original post by lazy_fish
No need to overcomplicate the question. Just think of sequences that would satisfy the requirements stated in the question.

Hope that helps.


Yes I thought I may have been overcomplicating it, but I'm not sure how I would be able to think of sequences that satisfy it?
Reply 3
Original post by _JC95
Yes I thought I may have been overcomplicating it, but I'm not sure how I would be able to think of sequences that satisfy it?


Well what sort of sequences do you know that obey the conditions in the question?
Reply 4
Original post by davros
Well what sort of sequences do you know that obey the conditions in the question?


Well sequences such as 1/n and 1/(2^n-1) would obey the conditions of a_n and b_n tending to 0..
Reply 5
Original post by _JC95
Well sequences such as 1/n and 1/(2^n-1) would obey the conditions of a_n and b_n tending to 0..


Good start!

Now what would happen if a_n and b_n were both 1/n? Would they both satisfy the conditions required? What would be the limit of a_n/b_n then? Does this give you any ideas?
Reply 6
Original post by davros
Good start!

Now what would happen if a_n and b_n were both 1/n? Would they both satisfy the conditions required? What would be the limit of a_n/b_n then? Does this give you any ideas?


Ahh I see, could I then use a_n= -3/n and b_n=1/n?
Reply 7
Original post by _JC95
Ahh I see, could I then use a_n= -3/n and b_n=1/n?


That's what I was thinking :smile:
Reply 8
Original post by davros
That's what I was thinking :smile:


Thanks :smile: I also have to think about, using the same conditions, examples of a_n and b_n that give a_n/b_n tends to +infinity and also the sequence a_n/b_n diverges
Reply 9
Original post by _JC95
Thanks :smile: I also have to think about, using the same conditions, examples of a_n and b_n that give a_n/b_n tends to +infinity and also the sequence a_n/b_n diverges


I don't want to give too much away because you'll benefit from thinking about this yourself, but think about the next most complicated thing after 1/n (e.g. powers).

By "diverges" do you mean doesn't tend to a limit as opposed to tending to infinity? (Some texts would include "tending to infinity" as diverging!)
Reply 10
Original post by davros
I don't want to give too much away because you'll benefit from thinking about this yourself, but think about the next most complicated thing after 1/n (e.g. powers).

By "diverges" do you mean doesn't tend to a limit as opposed to tending to infinity? (Some texts would include "tending to infinity" as diverging!)


Yes I understand for the +infinity one now, and by diverges I mean it doesn't tend to a limit. I can think of sequences such as (-1)^n which have this property, but unsure on how to combine 2 null sequences as a_n/b_n to derive this property.
Reply 11
Original post by _JC95
Yes I understand for the +infinity one now, and by diverges I mean it doesn't tend to a limit. I can think of sequences such as (-1)^n which have this property, but unsure on how to combine 2 null sequences as a_n/b_n to derive this property.


I couldn't see the answer to this myself at first, but I think I can now!

What did we do when we were trying to get a limit of -3 for the ratio?
Reply 12
Original post by davros
I couldn't see the answer to this myself at first, but I think I can now!

What did we do when we were trying to get a limit of -3 for the ratio?


I have figured it out now anyway, thanks :smile:
Reply 13
Original post by _JC95
I have figured it out now anyway, thanks :smile:


Good stuff!

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