Chapter 2.4, Problem 66E

Learning theory. In $1917$, L. L. Thurstone, a pioneer in quantitative learning theory, proposed the rational function

$f\left(x\right)=\frac{a\left(x+c\right)}{\left(x+c\right)+b}$

to model the number of successful acts per unit time that a person could accomplish after $x$ practice sessions. Suppose that for a particular person enrolled in a typing class,

$f\left(x\right)=\frac{55\left(x+1\right)}{\left(x+8\right)}x\ge 0$

where $f\left(x\right)$ is the number of words per minute the person is able to type after $x$ weeks of lessons.

(A) What does $f\left(x\right)$ approach as $x$ increases?

(B) Sketch a graph of function $f$, including any vertical or horizontal asymptotes.