IITJEE 06 - Limits

for x→0+ sinx→0+ and (1/x)→∞
so (sinx)^(1/x) →0

please correct me. i dont know where i am going wrong

for (1/x)^sinx taking logarithm and applying l'hospitals rule

ln L = lim sinx.ln(1/x)

ln L=lim -(lnx)/cosecx

applying lhspital rule
ln L= lim tanx(sinx/x)
ln L= .... tan x=0

i had made a mistake earlier here forgot that ln L=0 not L itself
so L=1