To fully understand the nature of horizontal asymptotes, one must first explore vertical
asymptotes. Asymptotes are when the graph of a function approaches a certain number but never
reaches it. A complex rational expression is essentially a fraction, and a fraction is essentially division.
The nature of division is that dividing a certain constant by a large number returns a progressively
smaller value, whereas dividing the constant by a small number returns a progressively larger value.
However, in division, at no point does the return value equate zero or infinity. The domain of a
complex rational expression with a variable on the denominator and a constant on the numerator is all
numbers except infinity and zero. Because infinity is not considered to be part of the set of all real
numbers, the only exception that remains is zero. This means the asymptote lies when the complex
rational expression is a divide by zero, which happens when the denominator is set to zero. This makes
solving for the vertical asymptote as conceptually elegantly solving for the zeros of the denominator by
any means, be it simple algebra, or quadratics.
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