Determine the following limits. 


lim x→∞ x^4+7 / x^5+x^2−x

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).

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

Evaluate f(3) if lim x→3^− f(x)=5,lim x→3^+ f(x)=6, and f is right-continuous at x=3.

Determine the following limits. 

lim θ→∞ cos θ / θ²

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).

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

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$, determine $\underset{x\to a^{+}}{\lim }f\left(x\right)$, $\underset{x\to a^{-}}{\lim }f\left(x\right)$, and $\underset{x\to a}{\lim }f\left(x\right)$.

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

Determine the following limits.

lim x→3 1/ x − 3(1 /√x + 1 − 1/2)