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Found in: Page 632

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# Graph and discuss the continuity of the functionf(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.

See the step by step solution

## 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.