1. Limits and Continuity

Determine ${\lim }_{x\to \infty }f\left(x\right)$ and ${\lim }_{x\to -\infty }f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x}{20x+1}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{6x^2-9x+8}{3x^2+2}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{3x^3-7}{x^4+5x^2}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{1}{2x^4-\sqrt{4x^8-9x^4}}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{\sqrt[3]{x^6+8}}{4x^2+\sqrt{3x^4+1}}$

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=4x(3x-\sqrt{9x^2+1})$

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{x^2-3}{x+6}$

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

$f\left(x\right)=\frac{x^2-2x+5}{3x-2}$

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{x^2-2x+5}{3x-2}$

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of f (if any).

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$