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:

, then

Now if we define , we can say:

directly from

So that means

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:

John