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lim(xrarroo) x^(2)sin(log(e)sqrt(cos(pi)...

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

A

0

B

`-(pi^(2))/(2)`

C

`-(pi^(2))/(4)`

D

`-(pi^(2))/(8)`

Text Solution

Verified by Experts

The correct Answer is:
C
`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)`
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