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It's not even technically true if you allow for negative denominators that are close to zero. Assuming a positive numerator and a positive denominator, the value of the fraction increases without bound (approaches infinity) as the denominator approaches zero. If the denominator is negative, however, the value of the fraction decreases without bound (approaches negative infinity) as the denominator approaches zero. These can be stated much more succinctly using one-sided limit notation.

Quotes aren't needed around real in "real" numbers. Division by zero is also undefined for complex numbers.