Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.
$f(-1)=-2$, $f\left(1\right)=2$, $f\left(0\right)=0$, $\underset{x\to \infty }{\lim }f\left(x\right)=1$, $\underset{x\to -\infty }{\lim }f\left(x\right)=-1$

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {√x if x<4

3 if x=4; a=4

x+1 if x>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).

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

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).

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

Consider the position function s(t) =−16t^2+100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

Estimate the following limits using graphs or tables.

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