Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
h(x)=e^x(x+1)^3

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

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 colony of squirrels is given by $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

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).

$f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}$

The following table gives the position $s\(\left\)(t\(\right\))$ of an object moving along a line at time $t$. Determine the average velocities over the time intervals $\(\left\[\lbrack\)1,1.01\(\right\]\rbrack\)$, $\(\left\[\lbrack\)1,1.001\(\right\]\rbrack\)$, and $\(\left\]\lbrack\)1,1.0001^{}\(\right\).]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

Evaluate each limit and justify your answer. 

lim x→1 (x+5x / x+2)^4

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.