Now they must mean not to use L'H in the lim above, but we can done the following (squeeze or sandwich theorem)
log(k)/log(2k) <= log(1+k)/log(2+k) <= log(1+k)/log(1+k) = 1
We use that log(x) is an monotonically ascending function. But
log k/log 2k = 1/[log 2/log k + 1] --> 1/[0+1] = 1 and we're done.