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 $p\left(t\right)=\frac{3500t}{t+1}$.

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=\sin x$

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

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

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

Let $g\(\left\)(x\(\right\))=\(\begin{cases}\)x^2+x & \(\text{if }\)x<1\\ a & \(\text{if }\)x=1\\ 3x+5 & \(\text{if }\)x>1\(\end{cases}\)$

a. Determine the value of a for which $g$ is continuous from the left at $1$.

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→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

Determine the following limits.

lim x→∞ (x⁴ − 1) / (x⁵ + 2)