# Horizontal asymptote

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**Algebraic Analysis on Horizontal Asymptotes**Let's take an in-depth look at the reasoning behind each case of horizontal asymptotes:

**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: $y = 0$

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

**Case 3**:if: degree of numerator > degree of denominator

then: horizontal asymptote: NONE

$i.e. f(x) = \frac{ax^{5}+......}{bx^{3}+......}$→$NO\; horizontal\; asymptote$

**Graphing Rational Functions**Sketch each rational function by determining:

i) vertical asymptote.

ii) horizontal asymptotes

**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

- $a(x) = \frac{x - 9}{x + 9}$
- $b(x) = \frac{x^{2}-9}{x^{2}+9}$
- $c(x) = \frac{x^{2}+9}{x^{2}-9}$
- $d(x) = \frac{x+9}{x^{2}-9}$
- $e(x) = \frac{x+3}{x^{2}-9}$
- $f(x) = \frac{x^{2}+9}{x+9}$
- $g(x) = \frac{-x-9}{-x^{2}-9}$
- $h(x) = \frac{-x^{2}-9}{-x^{2}+9}$
- $i(x) = \frac{x^{2}-9}{x+3}$
- $j(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}$