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Evaluate lim(ntooo) cos(pisqrt(n^(2)+n))...

Evaluate `lim_(ntooo) cos(pisqrt(n^(2)+n))` when n is an integer.

Text Solution

Verified by Experts

The correct Answer is:
0
`L=underset(ntooo)limcos(pisqrt(n^(2)+n))`
`=underset(ntooo)lim(-1)^(n)cos(npi-pisqrt(n^(2)+n))`
`=underset(ntooo)lim(-1)^(n)cos(pi(n-sqrt(n^(2)+n)))`
`=(-1)^(n)underset(ntooo)limcos((-npi)/(n+sqrt(n^(2)+n)))`
`=(-1)^(n)underset(ntooo)limcos((pi)/(1+sqrt(1+(1)/(n))))`
`=(-1)^(n)"cos"(pi)/(2)`
=0
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