We can do this by finding a common denominator for the fractions in the numerator of the function. The common denominator is $(2+x) \cdot 2 = 2(2+x)$.
So, we have:
$$\frac{\frac{1}{2+x}-\frac{1}{2}}{x} = \frac{\frac{2}{2(2+x)}-\frac{2+x}{2(2+x)}}{x} =
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