We have to consider for x>0 only because the functions are defined for x>0 only.
I think the option (iv) that you have given is
. Since there was some confusion in that option, so I gave how I have understood it.
Option (iv) is incorrect because it is not true for x=1.
Option (iii) is incorrect because f(x) is not defined for x=1. So, option (iii) is not always equal to 1.
Option (ii) is incorrect because of the following reason:-
At x=1, f(x) tends to infinity. So, 1/f(x) tends to zero. Whereas, at x=1, g(x) is equal to one (it is not approximate one, but it's an exact one)
But option (i) is correct.
Hope it helps!!!
If you have any doubts, then you may ask!!!
And regarding your first question, please retype it using Insert Equation. The question is not clear. And please mention if there's a use of greatest integer function (if any).