evaluate (i) `underset(xtooo)limsec^(-1)((x)/(sinx))" and "(ii) underset(xtooo)limsec^(-1)((sinx)/(x)).`

Options

A. `-(1)/(2sqrt(2))`
B. `(1)/(2sqrt(2))`
C. `(1)/(sqrt(2))`
D. does not exist

Answer

A

Explanation

`underset(xtooo)limsec^(-1)((x)/(sinx))=sec^(-1)((oo)/(sinoo))`
`=sec^(-1)((oo)/("any value between "-1" to "1))`
`=sec^(-1)(+-oo)=(pi)/(2)`
`underset(xtooo)limsec^(-1)((sinx)/(x))=sec^(-1)((sinoo)/(oo))`
`=sec^(-1)(("any value between "-1" to "1)/(oo))`
`=sec^(-1)0` = not defined
Hence, (i) exists but (ii) does not exist.