Math 22 Asymptotes

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Vertical Asymptotes and Infinite Limits

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.