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).


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

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

a. lim x→−2^+ f(x)

Estimate the following limits using graphs or tables.

$\underset{h\to 0}{\lim }\frac{\ln (1+h)}{h}$

Estimate the following limits using graphs or tables.

lim x→1 9(√2x − x^4 −3√x) / 1 − x^3/4

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).

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

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).

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