`lim_(xrarr(pi^(-))/(2)) [1+(cosx)^(cosx)]^(2)=`

Options

A. Does not exist
B. 1
C. e
D. 4

Answer

D

Explanation

`L=underset(xrarr(pi^(-))/(2))(lim)[1+(cosx)^(cosx)]^(2)`
Let `y=underset(xrarr(pi^(-))/(2))(lim)(cosx)^(cosx)`
`therefore" "log(y)=underset(xrarr(pi^(-))/(2))(lim)(cosx)log cosx`
`therefore" "log(y)=underset(xrarr(pi^(-))/(2))(lim)(log(cosx))/(sec(x))(oo//oo)` Using L'Hospital's rule, we get
`" "log(y)=underset(xrarr(pi^(-))/(2))(lim)[-((1)/(cosx)xx(sinx)/(secxtanx))]`
`rArr" "log(y)=underset(xrarr(pi^(-))/(2))(lim)(-cosx)=0`
`therefore" "y=e^(0)=1`
`therefore" "L=(1+1)^(2)=2^(2)=4`