Evaluate `lim_(ntooo) (-1)^(n-1)sin(pisqrt(n^(2)+0.5n+1)),"where "nin N`

Explanation

L=`(-1)^(n-1)underset(ntooo)limsin(pisqrt(n^(2)+0.5n+1))`
`=underset(ntooo)lim(-1)^(n-1)(-1)^(n-1)sin(npi-pisqrt(n^(2)+n/2+1))`
`=underset(ntooo)limsinpi{{(n-sqrt(n^(2)+n/2+1))(n+sqrt(n^(2)+n/2+1)))/(n+sqrt(n^(2)+n/2+1))}`
`=underset(ntooo)limsinpi{(n^(2)-n^(2)-n/2-1)/(n(1+sqrt(1+1/2n+1/n^(2)))}}`
`=underset(ntooo)limsinpi{(-n/2-1)/(n(1+sqrt(1+(1)/(2n)-(1)/(n^(2))))}}`
`=underset(ntooo)limsinpi((-1/2-1/n)/(1+sqrt(1+(1)/(2n)+(1)/(n^(2)))))=sin(-(pi)/(4))=-(1)/(sqrt(2))`