Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

Graph and discuss the continuity of the function
\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{sinxy}}{{xy}}, if xy \ne 0\\1, if xy = 0\end{aligned} \right.\)

The given function \(f(x,y) = \left\{ \begin{aligned}{l}\frac{{\sin xy}}{{xy}}, if{\rm{ }}xy \ne 0\\1, if{\rm{ }}xy = 0\end{aligned} \right.\) is continuous at all points.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding limit of f:

\(\mathop {\lim }\limits_{xy \to 0} \frac{{\sin xy}}{{xy}} = \frac{{\mathop {\lim }\limits_{xy \to 0} \sin xy}}{{\mathop {\lim }\limits_{xy \to 0} xy}} = \frac{0}{0}\)

Since both numerator and denominator approaches 0, use L-Hospital rule.

Step 2: Derivative of numerator using chain rule:

\(\mathop {\lim }\limits_{xy \to 0} \frac{{\sin xy}}{{xy}} = \frac{{(\sin xy)'}}{{(xy)'}}\)

\( = \mathop {\lim }\limits_{xy \to 0} \frac{{\cos (xy)(xy)}}{{(xy)}}\)

\( = \mathop {\lim }\limits_{xy \to 0} \cos (xy)\)

\( = \cos 0\). Apply limit

\( = 1\). Since \(\cos 0 = 1\)

Step 3: Graph the function:

Since the top part of the piecewise function is continuous at all point other than on the \(x\)-axis or \(y\)-axis. It is enough to determine the behavior at the axis to determine the continuity.

Since the second part of the piecewise function is equal to the limit of the first part, the function is continuous at all points.

Use any computer software to draw the graph of the given function.

Hence, the function is continuous at all points.

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Graph and discuss the continuity of the function
\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{sinxy}}{{xy}}, if xy \ne 0\\1, if xy = 0\end{aligned} \right.\)

Step by Step Solution

Step 1: Finding limit of f:

Step 2: Derivative of numerator using chain rule:

Step 3: Graph the function:

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

Graph and discuss the continuity of the function\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{sinxy}}{{xy}}, if xy \ne 0\\1, if xy = 0\end{aligned} \right.\)

Step by Step Solution

Step 1: Finding limit of f:

Step 2: Derivative of numerator using chain rule:

Step 3: Graph the function:

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Graph and discuss the continuity of the function
\(f(x,y) = \left\{ \begin{aligned}{l}\frac{{sinxy}}{{xy}}, if xy \ne 0\\1, if xy = 0\end{aligned} \right.\)