Frederick is trying to perfect a lemonade recipe. He is varying the amount of sugar water he adds to see what tastes best, and he knows he wants to put some sugar water in the mixture. So far, his recipe calls for 8 cups of lemon juice, and $x$ cups of sugar water. Let $L(x)$ be the fraction of cups in the final mixture that is sugar water.

a) Find a formula for $L(x)$.
$L(x) =$

b) What domain makes sense for $L(x)$ in the context of the problem?

Domain:
A. $(8,\infty)$
B. $[0,\infty)$
C. $(-\infty ,\infty)$
D. $(0,\infty)$

c) Find and interpret all $x$ values such that $L(x)=0.5$.

$x$-values:
A. 8
B. 8 and 0
C. 8, 0, and 0.5
D. None

Interpretation:
A. This is how many cups of sugar water Frederick needs to use to get 0.5 cups of lemonade.
B. This is the number of cups of sugar water in the mixture at any point.
C. This is the number of cups of sugar water at which the ratio between cups of lemon juice and sugar water is exactly half.
D. This is the number of cups of lemonade Frederick is making using 0.5 cups of sugar water.

d) Evaluate and interpret $\displaystyle \lim_{x\to\infty} L(x)$.

$\displaystyle \lim_{x\to\infty} L(x) =$

A. As Frederick adds more and more lemon juice to the mixture, the lemonade is essentially all lemon juice.
B. As Frederick adds more and more sugar water to the mixture, he essentially makes only 1 cup of lemonade.
C. As Frederick makes more and more lemonade, he only really needs to use 1 cup of sugar water in the mixture.
D. As Frederick adds more and more sugar water to the mixture, the lemonade is essentially all sugar water.

Hint: