Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f(x, y)=\left\\{\begin{array}{cl}\frac{\sin (x y)}{x y}, & \text { if } x y \neq 0 \\ 1, & \text { if } x y=0\end{array}\right.\)

Step-by-Step Solution

Answer

The function is continuous and equals 1 at all values where \( xy = 0 \) or as \( xy \to 0 \) from non-zero points.

Step 1: Understand the Function

The function given is a piecewise function. It has two cases: \( f(x, y) = \frac{\sin(xy)}{xy} \) when \( xy eq 0 \) and \( f(x, y) = 1 \) when \( xy = 0 \). Our goal is to evaluate and understand this function's behavior, especially at points where \( xy \to 0 \).

Step 2: Evaluate the Function for xy \neq 0

When \( xy eq 0 \), the function is given by \( f(x, y) = \frac{\sin(xy)}{xy} \). This is a standard form that approaches 1 as \( xy \to 0 \) due to the small angle approximation, where \( \sin(t) \approx t \) for values of \( t \) close to zero.

Step 3: Evaluate the Function for xy = 0

When \( xy = 0 \), we directly use the second part of the piecewise function, which states \( f(x, y) = 1 \). This ensures the function is defined at these points and matches the limit as \( xy \to 0 \) from non-zero values.

Step 4: Confirm Continuity at xy = 0

To ensure continuity at \( xy = 0 \), we check if the limit of \( \frac{\sin(xy)}{xy} \) as \( xy \to 0 \) is equal to 1, which it is, due to limit properties of the sine function and L'Hôpital's Rule. Thus, the function is continuous at \( xy = 0 \).

Step 5: Combine Results

Since the function takes the value 1 both for \( xy = 0 \) and as \( xy \to 0 \) from non-zero points, it remains consistent across its domain. The function is well-defined and continuous everywhere.

Key Concepts

Piecewise Functions

In calculus, a piecewise function is a type of function that is defined by different expressions based on different parts of the domain. These functions are divided into pieces, and each piece applies to a certain condition or interval. This type of function is useful because it allows us to model situations where a rule might change depending on the value.
Our exercise presents a piecewise function with two parts:

• For the case where \( xy eq 0 \), the function \( f(x, y) \) is defined as \( \frac{\sin(xy)}{xy} \).
• When \( xy = 0 \), the function \( f(x, y) \) takes the value 1.

Piecewise functions are common across different areas of mathematics and engineering because they can represent real-world scenarios where different conditions lead to different outcomes. Understanding how to handle each piece separately is essential in analyzing and graphing these functions.

Limits

Limits are a fundamental concept in calculus, central to understanding behavior near points the function does not directly cover. They allow us to analyze how functions behave when approaching a particular point from either direction on a coordinate plane. For our piecewise function, limits help us evaluate the behavior as \( xy \to 0 \).
To comprehend what happens to \( \frac{\sin(xy)}{xy} \) as \( xy \to 0 \), we utilize the limit definition: as \( t \to 0 \), \( \frac{\sin(t)}{t} \to 1 \). This approximation helps us understand part of the behavior of our function near this critical point.

• We confirm that as \( xy \to 0 \), the part of the function \( \frac{\sin(xy)}{xy} \) approaches 1, ensuring that the boundary condition fits smoothly.

Recognizing limits in piecewise functions is crucial for further analyzing their continuity and differentiability.

Continuity

Continuity in a function means there are no breaks, jumps, or holes at any point within the defined domain. A function is continuous at a point if the function’s value at that point is the same as the limit of the function as it approaches that point. In our context, we need to ensure continuity at \( xy = 0 \).
For our piecewise function, we need to check:

• The limit of \( \frac{\sin(xy)}{xy} \) as \( xy \rightarrow 0 \), which we determined to be 1.
• The value of the function at \( xy = 0 \), which is also defined as 1.

Since the limits from both directions match the function value defined at the point, the function is continuous at \( xy = 0 \). This ensures there is a smooth transition across all parts of the function's domain.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus for evaluating limits of indeterminate forms, particularly 0/0 or \(\infty/\infty\). It applies derivative calculus to simplify these ambiguous situations, making it easier to find limits.
Although our original piecewise function can be analyzed without applying L'Hôpital's Rule directly, understanding how it functions is beneficial. If our expression \( \frac{\sin(xy)}{xy} \) presented more complex behavior, we could consider differentiating the numerator and the denominator separately.

• L'Hôpital's Rule states that if \( \lim_{{x \to a}} \frac{f(x)}{g(x)} \) results in 0/0 or \(\infty/\infty\), then, under certain conditions, \( \lim_{{x \to a}} \frac{f'(x)}{g'(x)} \) can be evaluated instead.

Understanding L'Hôpital's Rule reinforces calculus techniques and prepares us to tackle more complex limit problems in the future.