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).

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

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

$f(-1)=-2$, $f\left(1\right)=2$, $f\left(0\right)=0$, $\underset{x\to \infty }{\lim }f\left(x\right)=1$, $\underset{x\to -\infty }{\lim }f\left(x\right)=-1$

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 the following limits.

$\underset{t\to \infty }{\lim }\frac{\cos \left(t\right)}{e^{3t}}$

Evaluate each limit and justify your answer. 

lim t→4 t−4 /√t−2