Find the limit of the sequence: sqrt(n) - sqrt(n^2 - 1)

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MarkSA

Junior Member
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Sep 8, 2007
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243
Hello,

Find the limit of the sequence: sqrt(n) - sqrt(n^2 - 1)

I am not sure where to begin with this problem. It is currently in indetermine form.. infinity - infinity I believe. I tried multiplying by the conjugate.
This got me:
[n - n^2 + 1]/[sqrt(n) + sqrt(n^2 - 1)]

I thought about dividing top and bottom by one over the largest power in the denominator. that's 1/n in this case I think. But I don't think that will accomplish much since I have a n^2 in the numerator. Any ideas?
 
You have the right idea. You just don't seem to know what to do with it. I'm relatively sure you mean to divide numerator and denominator by 'n'. Go ahead and do that, then interpret the results.

Hint: If you are finding the limit of n/1, what's the limit as n increases without bound?
 
MarkSA said:
Hello,

Find the limit of the sequence: sqrt(n) - sqrt(n^2 - 1)

I am not sure where to begin with this problem. It is currently in indetermine form.. infinity - infinity I believe. I tried multiplying by the conjugate.
This got me:
[n - n^2 + 1]/[sqrt(n) + sqrt(n^2 - 1)]

I thought about dividing top and bottom by one over the largest power in the denominator. that's 1/n in this case I think. But I don't think that will accomplish much since I have a n^2 in the numerator. Any ideas?
Click to expand...
..........................................
n - n^2 + 1
---------------------------
sqrt(n) + sqrt(n^2 - 1)

How about dividing by the largest power, period? That would be n^2. Advantage -- you get lots of things like 1/n^a, which go to zero. When the powers go inside the sqrt(), they double, of course:

1/n - 1 + 1/n^2
-----------------------------------
sqrt(n/n^4) + sqrt(n^2/n^4 - 1/n^4)

1/n - 1 + 1/n^2 0 - 1 + 0
--------------------------------------------- --> ------------- --> minus infinity?
sqrt(1/n^3) + sqrt(1/n^2 - 1/n^4) 0 + 0

I think that will do it for you. Actually, your idea would work, too, I think. You get to:

1 - n + 1/n 1 - n + 0
--------------------------------------------- --> ------------- --> minus infinity.
sqrt(1/n) + sqrt(1 - 1/n^2) 1 + 0
 
Thanks for the replies. I guess I wasn't really sure if I was doing it correctly.
 
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