Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.6 Calculating Limits Using the Limit Laws - 1.6 Exercises: 58

Answer

please see step-by-step

Work Step by Step

In exercise 57, we found that for a polynomial $p(x)$, $\displaystyle \lim_{x\rightarrow a}p(x)=p(a) \qquad (*)$ r is a rational function, $r(x)=\displaystyle \frac{p(x)}{q(x)}$ $p(x)$ and $q(x)$ are polynomials. Also, suppose that $q(a)\neq 0$. $\displaystyle \lim_{x\rightarrow a}r(x)=\lim_{x\rightarrow a}\frac{p(x)}{q(x)}$= ... Law 5, The limit of a quotient... $=\displaystyle \frac{\lim_{x\rightarrow a}p(x)}{\lim_{x\rightarrow a}q(x)}$ ... by the result of Exercise 57, (*) $=\displaystyle \frac{p(a)}{q(a)}$ $=r(a)$ so, $\displaystyle \lim_{x\rightarrow a}r(x)=r(a)$
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