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Evaluate underset(xto1)lim"sec"(pi)/(2^(...

Evaluate `underset(xto1)lim"sec"(pi)/(2^(x)).log_(e)x`.

A

`(pi)/(2)`

B

`-(2)/(pi ln 2)`

C

`(2)/(pi ln 2)`

D

`(pi)/(ln 2)`

Text Solution

Verified by Experts

The correct Answer is:
C
`lim_( x to 1) "sec"(pi)/(2^(x)) *ln x= lim_(x to 1) (ln)/("cos"(pi)/(2^(x)))=lim_(x to 0) ((1)/(x))/((-)pi "sin"(pi)/(2^(x)) *(1)/(2^(x)) ln 2(-1)) = (1)/(pi"sin"(pi)/(2) xx (1)/(2) xx ln 2)=(2)/(pi ln 2)`
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