# Horizontal asymptote

### Horizontal asymptote

#### Lessons

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 (x-axis)

$i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......}$→ 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.
Introduction to Horizontal Asymptote

• Horizontal Asymptotes define the right-end and left-end behaviors on the graph of a function.

• 3 cases of horizontal asymptotes in a nutshell…

• 2.
Algebraic Analysis on Horizontal Asymptotes

Let’s take an in-depth look at the reasoning behind each case of horizontal asymptotes:

a)
Case 1:

if: degree of numerator < degree of denominator

then: horizontal asymptote: y = 0 (x-axis)

$i.e. f(x) = \frac{ax^{3}+......}{bx^{5}+......}$→ horizontal asymptote: $y = 0$

b)
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: $y = \frac{a}{b}$

c)
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$

• 3.
Graphing Rational Functions

Sketch each rational function by determining:

i) vertical asymptote.

ii) horizontal asymptotes

a)
$f\left( x \right) = \frac{5}{{2x + 10}}$

b)
$g\left( x \right) = \frac{{5{x^2} - 13x + 6}}{{ - 2{x^2} + 3x + 2}}$

c)
$h\left( x \right) = \frac{{{x^3}}}{{20x - 100}}$

• 4.
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

a)
$a(x) = \frac{x - 9}{x + 9}$

b)
$b(x) = \frac{x^{2}-9}{x^{2}+9}$

c)
$c(x) = \frac{x^{2}+9}{x^{2}-9}$

d)
$d(x) = \frac{x+9}{x^{2}-9}$

e)
$e(x) = \frac{x+3}{x^{2}-9}$

f)
$f(x) = \frac{x^{2}+9}{x+9}$

g)
$g(x) = \frac{-x-9}{-x^{2}-9}$

h)
$h(x) = \frac{-x^{2}-9}{-x^{2}+9}$

i)
$i(x) = \frac{x^{2}-9}{x+3}$

j)
$j(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}$