Question

In Exercises \(33-38,\) use a computer algebra system to graph the function and find \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\) (if it exists). \(f(x, y)=\frac{2 x y}{x^{2}+y^{2}+1}\)

Short answer

Therefore, the limit of the function as \(x, y\) approach 0 is 0.

Step-by-step solution

Understanding the Function

The function given in this problem to analyze is \(f(x, y)=\frac{2 x y}{x^{2}+y^{2}+1}\). It is a rational function, where the numerator is a product of \(x, y\) and the denominator is a sum of squares \(x^{2}, y^{2}\) and unity.

Graphing the function

Use a computer algebra system to plot a 3-dimensional graph of the function \(f(x, y)=\frac{2 x y}{x^{2}+y^{2}+1}\). This would help understand the behavior of the function in the x-y plane.

Evaluating the Limit

To find the limit of the function as \(x, y\) approach \(0, 0\), which is written as \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)\), substitute \(x=0\) and \(y=0\) into the function. This gives \(f(0,0)=\frac{2 * 0 * 0}{0 + 0 + 1}\).

Solving for the Limit

After the substitution in step 3, the function simplifies to \(f(0,0)=0\). Therefore, \(\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0\).

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is a software tool that mathematics students and professionals use to solve complex mathematical problems and visualize various mathematical concepts. Having a powerful program like a CAS can greatly assist in graphing complicated functions, such as the rational function in our exercise, providing insight into their behavior and properties.

A CAS can handle a range of operations such as algebraic manipulations, calculus tasks like differentiation and integration, and even more complicated applications, including finding limits of multivariable functions. By entering the rational function \(f(x, y)=\frac{2 x y}{x^{2}+y^{2}+1}\) into a CAS, it can quickly generate the 3-dimensional graph and also compute the limit without the need for manual plotting or calculation. This not only saves time but also reduces the margin for error, making it a reliable resource for students working on their homework or researchers conducting complex analyses.

Utilizing a CAS can also be educational, as it can be used to verify manual calculations, allowing students to check their work against a dependable source.

Rational Function

A rational function is a function represented by the ratio of two polynomials. In the function \(f(x, y)=\frac{2 x y}{x^{2}+y^{2}+1}\), the numerator and the denominator are both polynomials, with the numerator being \(2xy\) and the denominator being a sum of squares and a constant \(x^{2}+y^{2}+1\). Rational functions are interesting as they often feature asymptotic behavior, discontinuities, and interesting limits, especially in a multivariable context.

To fully understand these functions, it is important to analyze how they behave as the variables approach certain values. For example, the limit of \(f(x, y)\) as \(x\) and \(y\) approach zero is pivotal in understanding the overall behavior around that point in space. In this case, substituting zero for both \(x\) and \(y\) simplifies the function substantially, as demonstrated in the exercise solution, ultimately leading to the function's limit as zero at that point. This is important not just for homework, but deeply understanding rational functions can be essential in fields such as engineering, physics, and economics, where they describe various real-world behaviors.

3-Dimensional Graph

Graphing in three dimensions provides a visual representation of multivariable functions that can offer significant insights into their nature and interactions. With the function \(f(x, y)\) given in our example, graphing it in 3D allows us to see the surface created by the function over the x-y plane.

The 3-dimensional graph will typically show contours, peaks, valleys, and planes, representing different values of the function at each point \( (x, y) \). When approaching the limit as both \(x\) and \(y\) go to zero, the graph enables us to visualize if the function approaches a specific value or behaves erratically. In the case of \(f(x, y)=\frac{2 x y}{x^{2}+y^{2}+1}\), the graph would help illustrate how the function levels off as it nears the origin, supporting the conclusion that the limit is zero.

Understanding 3-dimensional graphs is essential for students in fields that require visualizing complex systems or multidimensional data. It is one thing to calculate a limit or evaluate a function at a point algebraically, but viewing it on a 3-dimensional plot brings the concept to life and can greatly enhance comprehension.