Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.7 Limits at Infinity - Exercises - Page 83: 33


(a)$A$ (b)$A/2$ (c) $3.75~millimolars$

Work Step by Step

We are given the function $$R(s)=\frac{A s}{K+s}$$ (a) We have the limit \begin{align*} \lim _{s \rightarrow \infty} R(s)&=\lim _{s \rightarrow \infty} \frac{A s}{K+s}\\ &=\lim _{s \rightarrow \infty} \frac{A}{1+\frac{K}{s}}\\ &=A \end{align*} (b) We find the value of $R$ at $s=K$: \begin{align*} R(K)&=\frac{A K}{K+K}\\ &=\frac{A K}{2 K}\\ &=\frac{A}{2} \end{align*} Then the reaction rate $R(s)$ attains one-half of the limiting value A when $s = K$ (c) Since the limiting value is $0.1$, then \begin{align*} R(s)&=\frac{0.1 s}{1.25+s}\\ &=0.075 \end{align*} Hence $$s=\frac{(1.25)(0.075)}{0.025}=3.75$$
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