Compare and discuss the long-run behaviors of the following functions:

$$f(x)=\frac{x^2+1}{x^2+5}, \quad g(x)=\frac{x^3+1}{x^2+5}, \quad h(x)=\frac{x+1}{x^2+5} .$$

Typically, the average cost of production (as defined in the previous problem) decreases as the level of production increases. Is this always the case for the goods whose total cost function is graphed in the given figure? Use the result of the previous problem and explain your reasoning.

Find

(a) $\lim\limits_{x \rightarrow \infty} \dfrac{x\left(x^2-4\right)}{5+5 x^3}$

(b) $\lim\limits_{x \rightarrow-\infty} \dfrac{3 x(x-1)(x-2)}{5-6 x^4}$

Find a rational function of the form R(x) = 1/(1+cx) whose values equal those of $f(x)=e^x$ at $x=0$ and $x=1$. Plot and compare your approximation on the interval $(0,1)$ and comment on the result.

In the following exercise, find a power function through the two points.

(1,5)(3,27)

In this problem, give a possible formula for the polynomial.

(-1,0), (-1/2, -27/16), (0,0), (1,0)

It costs a company $30,000 to begin production of a good, plus$3 for every unit of the good produced. Let x be the number of units produced by the company.

(a) Find a formula for $C(x)$, the total cost for the production of $x$ units of the good.

(b) Find a formula for the company's average cost per unit, $a(x)$.

(c) Graph $y=a(x)$ for $0<x \leq 50,000,0 \leq y \leq 10$. Label the horizontal asymptote.

(d) Explain in economic terms why the graph of $a$ has the long-run behavior that it does.

(e) Explain in economic terms why the graph of $a$ has the vertical asymptote that it does.

(f) Find a formula for $a^{-1}(y)$. Give an economic interpretation of $a^{-1}(y)$.

(g) The company makes a profit if the average cost of its good is less than $\$ 5$ per unit. Find the minimum number of units the company can produce and make a profit.

For the following exercise, perform the operations. Express answers in reduced form.

$$\dfrac{x^3}{x-4} \bigg/ \dfrac{x^2}{x^2-2 x-8}$$

The table below gives approximate values for three functions, f, g, and h. One is exponential, one is trigonometric, and one is a power function. Determine which is which and find possible formulas for each.

An alcohol solution consists of 5 gallons of pure water and x gallons of alcohol, x>0. Let f(x) be the ratio of the volume of alcohol to the total volume of liquid. [Note that $f(x)$ is the concentration of the alcohol in the solution.]

(a) Find a possible formula for $f(x)$.

(b) Evaluate and interpret $f(7)$ in the context of the mixture.

(c) What is the zero of $f$ ? Interpret your result in the context of the mixture.

(d) Find an equation for the horizontal asymptote of $f$. Explain its significance in the context of the mixture.

What is the long-run behavior of the functions in the following problem?

$$y=\frac{2^x+3}{x^2+5}$$

In the following exercise, state the leading term.

$$x^2-3 x^3+5 x+7 x^3$$

For the following problem, find the real zeros (if any) of the polynomials.

$$y=x^4+6 x^2+9$$

Is the statement in the following problem true or false? Give an explanation for your answer.

The function $f(x)=\frac{1}{x}$ is a rational function.