You can write the function as:
#lnx/cscx = lnx sinx = (x lnx) (sinx/x)#
so that:
#lim_(x->0^+) lnx/cscx = (lim_(x->0^+) (x lnx))* ( lim_(x->0^+)(sinx/x))#
Both limits are well known, but we can remind how they can be calculated:
#lim_(x->0^+)(sinx/x)#
is in the indeterminate form #0/0# and can be calculated using l'Hospital's rule:
#lim_(x->0^+)(sinx/x) = lim_(x->0^+) (d/(dx) sinx)/(d/(dx) x) = lim_(x->0^+) cosx / 1 = 1#
#lim_(x->0^+) (x lnx)#
is in the indeterminate form #0/oo# but can be reduced to a different form by expressing it as a quotient:
#lim_(x->0^+) (x lnx) = lim_(x->0^+) lnx/(1/x)#
This is now in the form #oo/oo# and we apply again l'Hospital's rule:
# lim_(x->0^+) lnx/(1/x) = lim_(x->0^+) (d/(dx) lnx)/(d/(dx) (1/x)) = lim_(x->0^+) (1/x) / (-1/x^2) = lim_(x->0^+) (-x) = 0#
We can then conclude that:
#lim_(x->0^+) lnx/cscx = 0*1 = 0#
