Fractional rational functions
Tell a pole from a removable gap, read limits, find asymptotes and build a term from properties.
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Guide
A fractional rational function is a quotient of two polynomials. The denominator must not be zero: those values are not in the domain D.
If the denominator is zero at a point, check the numerator. If it is zero there too, cancel the linear factor: a removable gap remains. If the numerator is not zero, there is a pole and the graph approaches the vertical line.
For x → ±∞ the graph approaches an asymptote. If the numerator degree is one more than the denominator degree, the asymptote is a slanted line; polynomial division gives it.