SOLUTION: please help me to find out
lim (n,inf) 8n^2*pi*sqrt(n^2+n+1)
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Question 182519: please help me to find out
lim (n,inf) 8n^2*pi*sqrt(n^2+n+1)
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The 
in the radicand increases without bound as n increases without bound because:
so
Same thing for the 
in the radicand because 
The 1 in the radicand remains 1 regardless of the value of n, so we can say:
so
But the limit of the sum is the sum of the limits, so if we say:
 = n^2 + n + 1)
, then
 = \infty + \infty + 1 = \infty)
Now if we define  = sqrt(x) = x^{1 \over 2})
, we can say:
 = \infty)
directly from
So that means
\to \infty} H(G(n)) = \infty)
The factor of 8:
so
The factor of 
:
so
And finally the factor of outside of the radicand:
As previously shown:
The limit of the product is the product of the limits, so:
(\lim_{n\to \infty}n^2)(\lim_{n\to \infty}\pi)(\lim_{n\to \infty}sqrt{n^2 + n + 1})\cr\cr\text{ } =(8)(\infty)(\pi)(\infty)=\infty)
John
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