Determine the following limits.
lim x→∞ (x⁴ − 1) / (x⁵ + 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).

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

Evaluate each limit and justify your answer. 

lim x→∞(2x+1x / x)^3

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

If a function f represents a system that varies in time, the existence of lim ${\(\displaystyle\[\lim\)_{t\(\rightarrow\]\infty\)}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a culture of tumor cells is given by $p\left(t\right)=\frac{3500t}{t+1}$.

Find polynomials p and q such that f=p/q is undefined at 1 and 2, but f has a vertical asymptote only at 2. Sketch a graph of your function.

Determine the following limits.

$\underset{x\to 0}{\lim }\frac{x^3-5x^2}{x^2}$