4.6 Limits at infinity and asymptotes  (Page 9/14)

$f\left(x\right)=\frac{x+1}{x^2+5x+4},a=\mathrm{-1}$

$f\left(x\right)=\frac{x}{x-2},a=2$

$f\left(x\right)={\left(x+2\right)}^{3\text{/}2},a=\mathrm{-2}$

$f\left(x\right)={\left(x-1\right)}^{\mathrm{-1}\text{/}3},a=1$

$f\left(x\right)=1+x^{\mathrm{-2}\text{/}5},a=1$

$\underset{x\to \infty }{\text{lim}}\frac{1}{3x+6}$

$\underset{x\to \infty }{\text{lim}}\frac{2x-5}{4x}$

$\underset{x\to \infty }{\text{lim}}\frac{x^2-2x+5}{x+2}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{3x^3-2x}{x^2+2x+8}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{x^4-4x^3+1}{2-2x^2-7x^4}$

$\underset{x\to \infty }{\text{lim}}\frac{3x}{\sqrt{x^2+1}}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{\sqrt{4x^2-1}}{x+2}$

$\underset{x\to \infty }{\text{lim}}\frac{4x}{\sqrt{x^2-1}}$

$\underset{x\to \text{−}\infty }{\text{lim}}\frac{4x}{\sqrt{x^2-1}}$

$\underset{x\to \infty }{\text{lim}}\frac{2\sqrt{x}}{x-\sqrt{x}+1}$

$f\left(x\right)=x-\frac{9}{x}$

$f\left(x\right)=\frac{1}{1-x^2}$

$f\left(x\right)=\frac{x^3}{4-x^2}$

$f\left(x\right)=\frac{x^2+3}{x^2+1}$

$f\left(x\right)=\text{sin}\left(x\right)\text{sin}\left(2x\right)$

$f\left(x\right)=\text{cos}x+\text{cos}\left(3x\right)+\text{cos}\left(5x\right)$

$f\left(x\right)=\frac{x\text{sin}\left(x\right)}{x^2-1}$

$f\left(x\right)=\frac{x}{\text{sin}\left(x\right)}$

$f\left(x\right)=\frac{1}{x^3+x^2}$

$f\left(x\right)=\frac{1}{x-1}-2x$

$f\left(x\right)=\frac{x^3+1}{x^3-1}$

$f\left(x\right)=\frac{\text{sin}x+\text{cos}x}{\text{sin}x-\text{cos}x}$

$f\left(x\right)=x-\text{sin}x$

$f\left(x\right)=\frac{1}{x}-\sqrt{x}$

$x=1$ and $y=2$

$x=1$ and $y=0$

$y=4,$ $x=\mathrm{-1}$

$x=0$

[T] $f\left(x\right)=\frac{1}{x+10}$

[T] $f\left(x\right)=\frac{x+1}{x^2+7x+6}$

[T] $\underset{x\to \text{−}\infty }{\text{lim}}{x}^2+10x+25$

[T] $\underset{x\to \text{−}\infty }{\text{lim}}\frac{x+2}{x^2+7x+6}$

[T] $\underset{x\to \infty }{\text{lim}}\frac{3x+2}{x+5}$

$y=3x^2+2x+4$

$y=x^3-3x^2+4$

$y=\frac{2x+1}{x^2+6x+5}$

$y=\frac{x^3+4x^2+3x}{3x+9}$

$y=\frac{x^2+x-2}{x^2-3x-4}$

$y=\sqrt{x^2-5x+4}$

$y=2x\sqrt{16-x^2}$

$y=\frac{\text{cos}x}{x},$ on $x=\left[\mathrm{-2}\pi ,2\pi \right]$

$y=e^x-x^3$

$y=x\text{tan}x,x=\left[\text{−}\pi ,\pi \right]$

$y=x\text{ln}\left(x\right),x>0$

$y=x^2\text{sin}\left(x\right),x=\left[\mathrm{-2}\pi ,2\pi \right]$

For $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ to have an asymptote at $y=2$ then the polynomials $P\left(x\right)$ and $Q\left(x\right)$ must have what relation?

For $f\left(x\right)=\frac{P\left(x\right)}{Q\left(x\right)}$ to have an asymptote at $x=0,$ then the polynomials $P\left(x\right)$ and $Q\left(x\right).$ must have what relation?

If $f^{\prime }\left(x\right)$ has asymptotes at $y=3$ and $x=1,$ then $f\left(x\right)$ has what asymptotes?

Both $f\left(x\right)=\frac{1}{\left(x-1\right)}$ and $g\left(x\right)=\frac{1}{{\left(x-1\right)}^2}$ have asymptotes at $x=1$ and $y=0.$ What is the most obvious difference between these two functions?

True or false: Every ratio of polynomials has vertical asymptotes.