Horizontal asymptote  Rational Functions
Horizontal asymptote
Related concepts:
 Limits at infinity  horizontal asymptotes
Lessons
Notes:
There are 3 cases to consider when determining horizontal asymptotes:
1) Case 1:
if: degree of numerator < degree of denominator
then: horizontal asymptote: y = 0 (xaxis)
$i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......}$→ horizontal asymptote: $y = 0$
2) Case 2:
if: degree of numerator = degree of denominator
then: horizontal asymptote: y = $\frac{leading\; coefficient \;of\; numerator}{leading\; coefficient\; of\; denominator}$
$i.e. f(x) = \frac{ax^{5}+......}{bx^{5}+......}$→ horizontal asymptote: $y = \frac{a}{b}$
3) Case 3:
if: degree of numerator > degree of denominator
then: horizontal asymptote: NONE
$i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......}$→$NO\; horizontal\; asymptote$

1.
Algebraic Analysis on Horizontal Asymptotes
Let’s take an indepth look at the reasoning behind each case of horizontal asymptotes:

2.
Graphing Rational Functions
Sketch each rational function by determining:
i) vertical asymptote.
ii) horizontal asymptotes

3.
Identifying Characteristics of Rational Functions
Without sketching the graph, determine the following features for each rational function:
i) point of discontinuity
ii) vertical asymptote
iii) horizontal asymptote
iv) slant asymptote