Calculus (3rd Edition)

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

Chapter 15 - Differentiation in Several Variables - 15.2 Limits and Continuity in Several Variables - Exercises - Page 772: 25

Answer

By Squeeze Theorem we obtain the limit: $\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} \left( {{x^2} - 16} \right)\cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) = 0$

Work Step by Step

We have $ - 1 \le \cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) \le 1$. Suppose $x \ge 4$, so $ - \left( {{x^2} - 16} \right) \le \left( {{x^2} - 16} \right)\cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) \le {x^2} - 16$ Taking limits of the inequalities gives $\mathop { - \lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} {x^2} - 16 \le \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} \left( {{x^2} - 16} \right)\cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) \le \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} {x^2} - 16$ The two limits on both ends are equal to $0$. Now, if $x < 4$, then $\left( {{x^2} - 16} \right) \le \left( {{x^2} - 16} \right)\cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) \le - \left( {{x^2} - 16} \right)$ Taking limits of the inequalities gives $\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} {x^2} - 16 \le \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} \left( {{x^2} - 16} \right)\cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) \le \mathop { - \lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} {x^2} - 16$ In this case too, the two limits on both ends are equal to $0$. Therefore, by Squeeze Theorem we obtain the limit: $\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4,0} \right)} \left( {{x^2} - 16} \right)\cos \left( {\frac{1}{{{{\left( {x - 4} \right)}^2} + {y^2}}}} \right) = 0$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.