ASYMPTOTES OF RATIONAL FUNCTIONS
)(
)(
)( xD
xN
xfy
where N(x) and D(x) are polynomials
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By Joanna Gutt-Lehr, Pinnacle Learning Lab, last updated 1/2010
HORIZONTAL ASYMPTOTES, y = b
A horizontal asymptote is a horizontal line that is not part of a graph of a function but guides it for x-values
“far” to the right and/or “far” to the left. The graph may cross it but eventually, for large enough or small
enough values of x (approaching ), the graph would get closer and closer to the asymptote without touching
it. A horizontal asymptote is a special case of a slant asymptote.
A ”recipe” for finding a horizontal asymptote of a rational function:
Let deg N(x) = the degree of a numerator and deg D(x) = the degree of a denominator.
deg N(x) = deg D(x)
deg N(x) < deg D(x)
deg N(x) > deg D(x)
There is no horizontal asymptote.
Another way of finding a horizontal asymptote of a rational function:
Divide N(x) by D(x). If the quotient is constant, then y = this constant is the equation of a horizontal asymptote.
Examples
Ex. 1 Ex. 2
HA:
because approaches 0 as x increases.
HA :
because approaches 0 as x increases.
Ex. 3
= approaches as x increases (y = 3x – 3 is a slant asymptote.)
D(x) oft coefficien leading
)N(x oft coefficien leading
y
axis- x theis which 0y
