numerator by the denominator. We can use long division to do that: Notice that we don't need to finish the long division problem to find the remainder. We only need the terms that will make up the equation of the line. The slant asymptote is y − .
As you can see in this graph of the function, the curve approaches the slant asymptote y − but never crosses it: Example . Find the asymptotes of the curve f x ( ) = As lim →− + = −∞ and lim → + = ∞ Therefore x =− and x = are vertical asymptotes. As lim →∞ = lim →∞ = Fig. .
Asymptote Asymptote − − − − − = Fig. . – – – – y = x – x = y = x = – Fig. .
- - and lim →−∞ = lim →−∞ = Therefore, y = is a horizontal asymptote. This can also be obtained by synthetic division.