When x is not zero, it can be (x + 1/x).
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It depends on what you mean by an opposite. Every real number has an additive opposite but 0 does not have a multiplicative opposite.
Any positive numbers opposite is the same number just as a negative.
If the opposite is meant to be the additive opposite and not the multiplicative opposite, then their sum is zero. The reason is that is what defines an additive opposite!
The [multiplicative] opposite of -2 is -0.5, which is negative.
0 as the opposite of 3 is -3. 3+ (-3) = 0