`lim_(xrarroo) x^(2)sin(log_(e)sqrt(cos(pi)/(x)))`

Options

A. 0
B. `-(pi^(2))/(2)`
C. `-(pi^(2))/(4)`
D. `-(pi^(2))/(8)`

Answer

C

Explanation

`underset(xrarroo)(lim)x^(2)sin(log_(e)sqrt(cos.(pi)/(x)))" "("From "0 xx oo)`
`" "underset(xrarroo)(lim)(x^(2)sin(log_(e)sqrt(cos.(pi)/(x))))/((log_(e)sqrt(cos.(pi)/(x))))(log_(e)sqrt(cos.(pi)/(x)))`
`" "=underset(xrarroo)(lim)x^(2).log_(e)sqrt(cos.(pi)/(x))`
`" "=underset(xrarroo)(lim)(x^(2))/(2)log_(e)(cos.(pi)/(x))`
`" "=underset(xrarroo)(lim)(x^(2))/(2)(log_(e)[1+(cos.(pi)/(x)-1)])/((cos.(pi)/(x)-1))(-2sin^(2).(pi)/(2x))`
`" "=underset(xrarroo)(lim)(x^(2))/(2)((-2sin^(2).(pi)/(2x)))/(((pi)/(2x))^(2)).((pi)/(2x))^(2)`
`" "=-(pi^(2))/(4)`