Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.1 - Curves in Space and Their Tangents - Exercises 13.1 - Page 747: 31


$\lim\limits_{t \to t_0}r(t)=r(t_0)$ and $r(t)$ is continuous at $t=t_0$

Work Step by Step

Consider $\lim\limits_{t \to t_0}r(t)=\lt f(t), g(t), h(t) \gt$ Here, we have that when $r(t)$ is continuous at $t=t_0$ thus, $\lim\limits_{t \to t_0}r(t)=r(t_0)=\lt f(t_0), g(t_0), h(t_0) \gt$ This implies that $\lim\limits_{t \to t_0} f(t)=f(t_0) \\\lim\limits_{t \to t_0} g(t)=g(t_0)\\\lim\limits_{t \to t_0} h(t)=h(t_0)$ Therefore, $\lim\limits_{t \to t_0}r(t)=r(t_0)$ and $r(t)$ is continuous at $t=t_0$
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