Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - 1.4 Exercises: 77

Answer

$f(x)$ is continuous over the interval $(-\infty, \infty).$

Work Step by Step

Using Theorem $1.11:$ $f(x)=\dfrac{g(x)}{h(x)}\to g(x)=x$ and $h(x)=x^2+x+2.$ $f(x)$ is continuous as long as both $g(x)$ and $h(x)$ are continuous and $h(x)\ne0.$ $g(x)$ is continuous over the interval $(-\infty, \infty)$ and $h(x)$ is continuous over the interval $(-\infty, \infty)$; furthermore, $h(x)$ is never equal to zero since the determinant of the quadratic, $\Delta=(1)^2-4(1)(2)=-7\to$ no real roots. All this shows that $f(x)$ is continuous over the interval $(-\infty, \infty).$
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