Math 22 Asymptotes

Vertical Asymptotes and Infinite Limits

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

Example: Find the a vertical Asymptotes as below:

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

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

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

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

Definition of Horizontal Asymptote

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

Horizontal Asymptotes of Rational Functions

 Let $f(x)={\frac {p(x)}{q(x)}}$ 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.

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