In what ways can horizontal, vertical, and oblique asymptotes be identified?

A horizontal asymptote y = a is a horizontal line which a curve approaches as x approaches positive or negative infinity. If the limit of a curve as x approaches either positive or negative infinity is a, then y=a is a horizontal asymptote.

A vertical asymptote x = b is a vertical line that a curve approaches but never crosses. The value b is not in the domain of the curve. More precisely if the limit of a curve as x approaches b is either positive or negative infinity then x=b is a vertical asymptote.

An oblique asymptote is a diagonal line (a line whose slope is either positive or negative) that a curve approaches. For a rational function R(x) = P(x) / Q (x) an oblique asymptote y = my + b is obtained by dividing P(x) by Q (x). Doing so will yield a quotient and remainder. If we set the quotient equal to y that gives the equation of the oblique asymptote.