a trigonometric limit question

Hey! I was doing some practice problems and I stumbled upon two rather challenging questions that I cannot for some reason solve.

The questions correspond to letter F and G of number 3)

like i don't even know how to algebraically prove this. I had no problem with the other questions, but I am struggling with these two questions.

Note: My teacher said you cannot use derivatives in order to find the answer. so if possible do not use derivatives.
if possible please show me algebraically or through reasoning why the limit is what it is based on the question.

Thx in Advance

For f use sin(x-1) = sin(x)cos(1) -sin(1)cos(x)

For g

condider one sided limits as x -> 0+ |sin(x)| =sin(x)

and sqrt |sin(x)|/x = sqrt[sin(x)/x^2]

use L'Hopitals rule on the term inside the sqrt

Similarly approach the left hand limit

we are not allowed to use l'hopitals rule at this stage in time. the only way im allowed to do it is with trig identities and cancellation of common terms.

For lim sin(x)/x^2

You should have lim sin(x)/x = 1 as x ->0 at you disposal

lim sin(x)/x^2 = lim sin(x)/x lim 1/x = 1 * inf = inf