# Math 22 Asymptotes

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## Vertical Asymptotes and Infinite Limits

 If ${\displaystyle f(x)}$ approaches infinity (or negative infinity) as ${\displaystyle x}$ approaches ${\displaystyle c}$
from the right or from the left, then the line ${\displaystyle x=c}$ is a vertical asmptote of the graph of ${\displaystyle f}$



Example: Find the a vertical Asymptotes as below:

1) ${\displaystyle f(x)={\frac {x+3}{x^{2}-4}}}$

Solution:
Notice ${\displaystyle f(x){\frac {x+3}{x^{2}-4}}={\frac {x+3}{(x-2)(x+2)}}}$
Let the denominator equals to zero, ie: ${\displaystyle (x-2)(x+2)=0}$, hence ${\displaystyle x=-2}$ or ${\displaystyle x=2}$
Therefore, ${\displaystyle f(x)}$ has vertical asymptotes at ${\displaystyle x=2}$ and ${\displaystyle x=-2}$

2) ${\displaystyle f(x)={\frac {x^{2}-x-6}{x^{2}-9}}}$

Solution:
Notice ${\displaystyle f(x){\frac {x^{2}-x-6}{x^{2}-9}}={\frac {(x-3)(x+2)}{(x-3)(x+3)}}={\frac {x+2}{x+3}}}$
Let the denominator equals to zero, ie: ${\displaystyle (x+3)=0}$, hence ${\displaystyle x=-3}$
Therefore, ${\displaystyle f(x)}$ has vertical asymptote at ${\displaystyle x=-2}$

## Definition of Horizontal Asymptote

 If ${\displaystyle f}$ is a function and ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ are real numbers, then the statements
${\displaystyle \lim _{x\to \infty }f(x)=L_{1}}$ and ${\displaystyle \lim _{x\to -\infty }f(x)=L_{2}}$
denote limits at infinity. The line ${\displaystyle y=L_{1}}$ and ${\displaystyle y=L_{2}}$ are horizontal asymptotes
of the graph of ${\displaystyle f}$


## Horizontal Asymptotes of Rational Functions

 Let ${\displaystyle f(x)={\frac {p(x)}{q(x)}}}$ be a rational function.
1. If the degree of the numerator is less than the degree of the denominator,
then  is a horizontal asymptote of the graph of  (to the left and to the right).
2. If the degree of the numerator is equal to the degree of the denominator,
then  is a horizontal asymptote of the graph of  (to the left and to the right),
where  and  are the leading coefficients of  and , respectively.
3. If the degree of the numerator is greater than the degree of the denominator,
then the graph of  has no horizontal asymptote.