Limits AT Infinity: Rational Functions

$\underset{x+-}{lim}f(x)=\underset{x+-}{lim}\frac{p(x)}{q(x)}=\underset{x+-}{lim}\frac{\text{l}\text{e}\text{a}\text{d}\text{i}\text{n}\text{g}\text{t}\text{e}\text{r}\text{m}of\text{t}\text{o}\text{p}}{\text{l}\text{e}\text{a}\text{d}\text{i}\text{n}\text{g}\text{t}\text{e}\text{r}\text{m}of\text{b}\text{o}\text{t}\text{t}\text{o}\text{m}}$ Simplify and cancel common terms of top and bottom.

Three possibilities:

There are $x^{\text{'}}\text{'}$s left in bottom, but not top: $\underset{x+-}{lim}\frac{k}{x^p}=0($very important limit called the reciprocal limit$)$

There are no $x^{\text{'}}$ s left in bottom or top: $\underset{x+-}{lim}\frac{\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}\text{l}\text{e}\text{f}\text{t}on\text{t}\text{o}\text{p}}{\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}\text{l}\text{e}\text{f}\text{t}on\text{b}\text{o}\text{t}\text{t}\text{o}\text{m}}=\frac{\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}\text{l}\text{e}\text{f}\text{t}on\text{t}\text{o}\text{p}}{\text{n}\text{u}\text{m}\text{b}\text{e}\text{r}\text{l}\text{e}\text{f}\text{t}on\text{b}\text{o}\text{t}\text{t}\text{o}\text{m}}($limit of a constant$)$

There a x$\text{'}$s left on top, but not bottom: $\underset{x+-}{lim}\frac{x^p}{k}=$ solve like a polynomial limit $$ either $$ or $-$

Similar to PlayPosit video question$($s$)$ # $q,$ Evaluate the following limits$.$ USE PROPER LIMIT NOTATION.

a$.\underset{x+}{lim}\frac{x^5+3x-1}{x^9-4x+5}=$

b$.,\underset{x-}{lim}\frac{-2x^4-3x+1}{(x^2-5{)}^2}=$

c$.\underset{x-}{lim}\frac{5x^{11}-2x^4+7x-2}{x^7+6x-2}=$

Vertical and Horizontal Asymptotes $($using calculus and limits$)$

Horizontal Asymptote: Take the limit AT infinity of the function: $\underset{x+-}{lim}f(x)$

Vertical Asymptote: Find the $x$ values that make the denominator zero and take the limit of the function as $x$ approaches those numbers. Determine if there is a hole $(\frac{0}{0})$ or vertical asymptote $(\frac{n}{0}).$

Similar to PlayPos$$t video $$ question$($s$)$ # $q,.$

Find all horizontal and vertical asymptotes of $f(x)=\frac{x^2-2x-3}{x^2-1}$ using limits$.$ State if there are any holes in the graph, and if yes, give the coordinates. Failure to use limits for all asymptotes will result in zero credit.