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Definition of a Vertical Asymptote
Where the function is undefined at a certain x-value(s)
First Definition of a Horizontal Asymptote
If the degree of the numerator is greater than the degree of the denominator then there is no horizontal asymptote
Second definition of an HA
If the degree of the numerator is equal to the degree of the denominator then the HA is: y=coefficient/coefficient
Third definition of an HA
If the degree of the numerator is less than the degree of the denominator then the HA is y=0
Definition of a slant asymptote
If the degree of the numerator is bigger than the degree of the denominator by exactly 1, then you use SYNTHETIC DIVISION