Given a domain of #[0.01, 1]#, #1/x# has a range of #[1,100]#. Thus, the question is equivalent to asking how many times #sin(x)# crosses the #x#-axis on the interval #[1, 100]#.
As the graph of a function crosses the #x#-axis at the points where the function evaluates to #0#, and #sin(x) = 0 <=> x = npi, n in ZZ#, all that remains is to count the multiples of #pi# in the interval #[1,100]#
We can verify that #0 < 1 < pi# and #31pi < 100 < 32pi#, meaning the only integers in which #npi in [1, 100]# holds are #n=1, 2, ..., 31#. As there are #31# such values, #sin(x)=0# has #31# solutions on #[1, 100]#. As this is equivalent to our original problem, we have that #sin(1/x)# crosses the #x#-axis #31# times for #x in [0.01, 1]#
