Evaluating The Limit Of (1 - Cos(xy)) / Xy As (x, Y) Approaches (0, 0)

Introduction

In the fascinating realm of multivariable calculus, we often encounter limits that require careful evaluation. This article delves into the intriguing limit of the function (1 - cos(xy)) / xy as the point (x, y) approaches (0, 0). Understanding such limits is crucial for grasping the behavior of functions near specific points and forms a cornerstone of advanced calculus concepts. This exploration will not only solidify your understanding of limits but also enhance your problem-solving skills in calculus.

We will embark on a detailed analysis, employing various techniques and perspectives to unravel the intricacies of this limit. By examining the function's behavior along different paths and utilizing established limit theorems, we aim to provide a comprehensive understanding of its convergence. This journey will involve a blend of theoretical insights and practical applications, making it an enriching experience for anyone keen on mastering calculus.

Understanding the Limit

To begin, let's clearly define the limit we aim to evaluate:

$$\lim_{(x, y) \rightarrow (0, 0)} \frac{1 - \cos(xy)}{xy}$$

This limit represents the value that the function (1 - cos(xy)) / xy approaches as the point (x, y) gets arbitrarily close to the origin (0, 0). Evaluating this limit directly can be challenging due to the indeterminate form that arises when we substitute x = 0 and y = 0. The numerator becomes 1 - cos(0) = 0, and the denominator becomes 0, leading to a 0/0 indeterminate form. This necessitates the use of advanced techniques to resolve the limit.

The presence of trigonometric functions adds another layer of complexity. The behavior of cosine around 0 is well-known, but its interaction with the product xy requires careful consideration. We must explore different paths of approach to the origin to ensure the limit's existence and uniqueness. This involves analyzing the function's behavior along various curves and lines that converge to the origin.

The significance of this limit extends beyond a mere mathematical exercise. It touches upon fundamental concepts in calculus, such as continuity and differentiability. Understanding how such limits behave is essential for analyzing the behavior of more complex functions and for applications in fields like physics, engineering, and economics. Therefore, a thorough understanding of this limit is invaluable for anyone pursuing these disciplines.

Method 1: Using the Standard Limit

One effective approach to evaluating this limit is by leveraging a well-known trigonometric limit:

$$\lim_{u \rightarrow 0} \frac{1 - \cos(u)}{u} = 0$$

This standard limit is a cornerstone in calculus and can be derived using various methods, including L'Hôpital's Rule or geometric arguments. Our strategy involves transforming the given limit into a form where this standard limit can be directly applied. To achieve this, we make a strategic substitution.

Let's substitute u = xy. As (x, y) approaches (0, 0), the product xy also approaches 0. Therefore, we can rewrite the limit as:

$$\lim_{(x, y) \rightarrow (0, 0)} \frac{1 - \cos(xy)}{xy} = \lim_{u \rightarrow 0} \frac{1 - \cos(u)}{u}$$

Now, the limit is in the exact form of the standard limit we mentioned earlier. Applying the standard limit, we immediately find that:

$$\lim_{u \rightarrow 0} \frac{1 - \cos(u)}{u} = 0$$

Thus, we conclude that:

$$\lim_{(x, y) \rightarrow (0, 0)} \frac{1 - \cos(xy)}{xy} = 0$$

This method demonstrates the power of recognizing and applying standard limits. By making a clever substitution, we simplified the original limit into a manageable form. This technique is widely applicable in calculus and is a valuable tool in your problem-solving arsenal.

The beauty of this approach lies in its simplicity and directness. Once the substitution is made, the problem reduces to a straightforward application of a known result. This underscores the importance of mastering fundamental limits and being able to identify opportunities to apply them.

Method 2: L'Hôpital's Rule

Another powerful technique for evaluating limits of indeterminate forms is L'Hôpital's Rule. This rule is particularly useful when dealing with limits that result in forms like 0/0 or ∞/∞. To apply L'Hôpital's Rule, we need to ensure that the limit satisfies the necessary conditions, namely, that the numerator and denominator are differentiable and that the limit is indeed of an indeterminate form.

In our case, the limit is:

$$\lim_{(x, y) \rightarrow (0, 0)} \frac{1 - \cos(xy)}{xy}$$

As we established earlier, this limit results in the indeterminate form 0/0 when we directly substitute x = 0 and y = 0. This makes L'Hôpital's Rule a viable option.

However, L'Hôpital's Rule is typically applied to functions of a single variable. To adapt it to our multivariable limit, we can consider approaching (0, 0) along a specific path. Let's consider the path where y = kx, where k is a constant. This path represents a straight line passing through the origin.

Substituting y = kx into the limit, we get:

$$\lim_{x \rightarrow 0} \frac{1 - \cos(kx^2)}{kx^2}$$

Now we have a limit involving a single variable, x. We can apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to x:

$$\frac{d}{dx}(1 - \cos(kx^2)) = 2kx \sin(kx^2)$$

$$\frac{d}{dx}(kx^2) = 2kx$$

Applying L'Hôpital's Rule, the limit becomes:

$$\lim_{x \rightarrow 0} \frac{2kx \sin(kx^2)}{2kx} = \lim_{x \rightarrow 0} \sin(kx^2)$$

As x approaches 0, kx^2 also approaches 0, and sin(kx^2) approaches 0. Therefore, the limit along this path is 0.

It is crucial to note that applying L'Hôpital's Rule along one path does not guarantee that the limit exists and is the same along all paths. However, it provides strong evidence in this case. To further solidify our conclusion, we can explore other paths or employ the standard limit method discussed earlier.

This application of L'Hôpital's Rule showcases its adaptability to multivariable limits when combined with a strategic choice of path. It reinforces the importance of understanding and applying fundamental calculus techniques in diverse scenarios.

Method 3: Taylor Series Expansion

Another elegant and insightful method for evaluating limits, especially those involving trigonometric functions, is the use of Taylor series expansions. Taylor series provide a way to represent a function as an infinite sum of terms involving its derivatives at a specific point. This representation can be particularly useful in approximating the function's behavior near that point.

In our case, we are interested in the behavior of the function (1 - cos(xy)) / xy as (x, y) approaches (0, 0). The Taylor series expansion for cos(u) around u = 0 is given by:

$$\cos(u) = 1 - \frac{u^2}{2!} + \frac{u^4}{4!} - \frac{u^6}{6!} + ...$$

This series converges for all real values of u. Now, let's substitute u = xy into the expansion:

$$\cos(xy) = 1 - \frac{(xy)^2}{2!} + \frac{(xy)^4}{4!} - \frac{(xy)^6}{6!} + ...$$

Next, we substitute this expansion into our limit expression:

$$\frac{1 - \cos(xy)}{xy} = \frac{1 - (1 - \frac{(xy)^2}{2!} + \frac{(xy)^4}{4!} - \frac{(xy)^6}{6!} + ...)}{xy}$$

Simplifying the expression, we get:

$$\frac{1 - \cos(xy)}{xy} = \frac{\frac{(xy)^2}{2!} - \frac{(xy)^4}{4!} + \frac{(xy)^6}{6!} - ...}{xy}$$

Now, we can factor out xy from the numerator:

$$\frac{1 - \cos(xy)}{xy} = xy(\frac{1}{2!} - \frac{(xy)^2}{4!} + \frac{(xy)^4}{6!} - ...)$$

Dividing by xy, we obtain:

$$\frac{1 - \cos(xy)}{xy} = \frac{xy}{2!} - \frac{(xy)^3}{4!} + \frac{(xy)^5}{6!} - ...$$

Now, as (x, y) approaches (0, 0), all terms in the series approach 0. Therefore, the limit is:

$$\lim_{(x, y) \rightarrow (0, 0)} \frac{1 - \cos(xy)}{xy} = 0$$

This method beautifully illustrates the power of Taylor series in analyzing the behavior of functions near a point. By representing the cosine function as an infinite sum, we were able to simplify the limit and evaluate it directly. This approach is widely applicable in calculus and provides a deeper understanding of function approximation.

The Taylor series method not only confirms our earlier results but also provides additional insight into the function's behavior. It shows how the higher-order terms become negligible as (x, y) approaches (0, 0), leading to the limit's convergence to 0.

Conclusion

In this comprehensive exploration, we have successfully evaluated the limit of the function (1 - cos(xy)) / xy as (x, y) approaches (0, 0). We employed three distinct methods—the standard limit, L'Hôpital's Rule, and Taylor series expansion—each providing a unique perspective and reinforcing the same conclusion: the limit is 0.

The standard limit method demonstrated the power of recognizing and applying fundamental results. By making a clever substitution, we transformed the limit into a familiar form and obtained the solution directly. This approach underscores the importance of mastering basic limits and their applications.

L'Hôpital's Rule offered a different perspective, highlighting its adaptability to multivariable limits when combined with a strategic choice of path. While applying L'Hôpital's Rule along one path does not guarantee the limit's existence along all paths, it provided strong evidence in this case and showcased the technique's versatility.

The Taylor series expansion provided an elegant and insightful approach, revealing the underlying structure of the cosine function and its behavior near 0. This method not only confirmed our earlier results but also deepened our understanding of function approximation and the convergence of infinite series.

This exploration serves as a testament to the richness and interconnectedness of calculus concepts. By employing multiple methods, we not only solved the problem but also gained a more profound appreciation for the tools and techniques available to us. The limit we investigated is not merely a mathematical curiosity; it represents a fundamental concept in multivariable calculus with applications in various fields. Mastering such limits is crucial for anyone pursuing advanced studies in mathematics, physics, engineering, or related disciplines.

Furthermore, this exercise emphasizes the importance of a multifaceted approach to problem-solving. By having a diverse set of techniques at your disposal, you can tackle complex problems from different angles and gain a more complete understanding. This ability is invaluable in both academic and professional settings.

In conclusion, the limit of (1 - cos(xy)) / xy as (x, y) approaches (0, 0) is indeed 0. This exploration has not only demonstrated this result but also provided a comprehensive journey through various calculus techniques, enriching our understanding of limits and their applications. Keep exploring, keep questioning, and keep mastering the fascinating world of calculus!