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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Which of the following expressions equals exactly 1 for any non-zero real value of x?

I is true since the result of the division is the sign of x whose absolute value is always 1. If you do not understand this, here is a more detailed explanation:

Denote the sign of a number x with sgn(x).

This means that if x is negative then sgn(x)=-1 and if x is positive or zero, then sgn(x)=1.

Any number x can be written as: and in this notation the absolute value of it is separated from its sign.

Obviously, if we now divide x by its absolute value, only its sign remains. The absolute value of the sign is 1.

Note that to prove this identity it is not sufficient to take an example such as x=1.25 and claim that if it is true for this value then it is true for any non-zero real x.

II is not defined for x positive. Indeed, if x is positive, it is exactly equal to its absolute value, therefore the difference between the two is zero and we have a division by 0. When x is negative, the ratio is indeed 1: but, because it is not defined for any real value x, then II is false.

III is true since:  IV is false, since the floor of x is equal to x only when x is an integer. To prove that an identity is false, we only need to show there exists one counterexample. It is mathematically correct to say that if the expression is not 1 for some special value of x such as 1.25 then it is not true that it is 1 for any non-zero real x.

V is true since the factorial of zero is, by convention, equal to 1.