Question

In Exercises 29 and \(30,\) use a graphing utility to make a table showing the values of \(f(x, y)\) at the given points. Use the result to make a conjecture about the limit of \(f(x, y)\) as \((x, y) \rightarrow(0,0)\). Determine whether the limit exists analytically and discuss the continuity of the function. \(f(x, y)=\frac{y}{x^{2}+y^{2}}\) Path: \(y=0\) Points: (1,0) (0.5,0),(0.1,0) (0.01,0),(0.001,0) Path: \(y=x\) Points: (1,1) (0.5,0.5),(0.1,0.1) (0.01,0.01),(0.001,0.001)

Short answer

The limit of \(f(x, y)\) as \((x, y) \rightarrow(0,0)\) does not exist. Therefore, the function \(f(x, y)=\frac{y}{x^{2}+y^{2}}\) is not continuous at \((0,0)\).

Step-by-step solution

Evaluating function along the path \(y=0\)

Start by plugging in the points on the path \(y=0\) into the function. The function becomes \(f(x,y)=\frac{0}{x^{2}+0}\). Hence, the function at these points evaluates to \(0\), regardless of the value of \(x\).

Evaluating function along the path \(y=x\)

Next, plug in the points on the path \(y=x\) into the function. The function becomes \(f(x,y)=\frac{x}{x^{2}+x^{2}}\), which simplifies to \(f(x,y)=\frac{x}{2x^{2}}\), which further simplifies to \(f(x,y)=\frac{1}{2x}\). As \(x\) approaches \(0\), \(f(x,y)\) will undefined.

Making a conjecture about the limit

Based on the different values of the function when approached along the two different paths, it can be concluded that the limit as \((x, y) \rightarrow(0,0)\) does not exist. This is because for the limit to exist, the function should approach the same value regardless of the path taken to the limit, which is not the case here.

Determining the limit analytically and discussing the continuity of the function

The analytical determination of the limit confirms the conjecture. The limit does not exist as \((x, y) \rightarrow(0,0)\). Therefore, the function is not continuous at the point \((0,0)\). A function is continuous when the limit exists and the function is defined at that point. In this case, while the function is defined for \((x,y)=(0,0)\), the limit at this point does not exist, so the function is not continuous at \((0,0)\).

Key concepts

Continuity in Multivariable Calculus

Understanding continuity in the context of multivariable calculus is crucial when analyzing functions with more than one variable. Continuity at a point in multivariable calculus follows a similar definition to that in single-variable calculus—a function is continuous at a point if the limit exists at that point and is equal to the function's value there. For a function of two variables, like in our exercise, continuity at the point \( (x_0, y_0) \) requires that no matter what path you take towards \( (x_0, y_0) \), the function's value must converge to a single number, which should be the function's value at \( (x_0, y_0) \).

In the given exercise, the function \( f(x, y)=\frac{y}{x^{2}+y^{2}} \) is not continuous at \( (0,0) \) since the limit approached from different paths yields different results. In simpler terms, the destination is the same, but the route you take to get there changes the outcome—signaling that the function has a discontinuity at that point.

Evaluating Limits Analytically

Evaluating limits in multivariable calculus often requires consideration of multiple approaches to the point of interest. If these approaches yield different results, the limit does not exist. Analytically evaluating limits involves algebraically manipulating the function to resolve indeterminacies and applying limit laws.

In our exercise, two different paths give us different limits, demonstrating that no single limit exists for \( (x, y) \) approaching \( (0,0) \). Analytically, a limit in multivariable calculus is proven to exist only if every possible path to \( (x_0, y_0) \) results in the same limit value. When this doesn't happen, as with our function where one path gives us 0 and another becomes undefined, we assert that the limit fails to exist.

Graphing Utility for Functions

Graphing utilities provide visual representations of functions, and these visual aids are especially beneficial in multivariable calculus where grasping the function's behavior can be more complex. Graphs help in visualizing paths and the function's values at various points, thereby aiding in understanding limits and continuity.

Through graphing utilities, we can plot the function's surface or contours, which may hint at the function's limits when approaching certain points from different directions. In the exercise, using a graphing utility to visualize \( f(x, y) \) can aid in making an educated conjecture about the limit as \( (x, y) \) approaches \( (0,0) \), signaling discontinuities if the surface or contours appear disjointed or behave erratically near that point.

Multivariable Calculus

Multivariable calculus is the branch of calculus that extends the concepts of differentiation and integration to functions with multiple variables. It deals with functions like \( f(x_1, x_2, ..., x_n) \), which have more than one input. In addition to studying limits and continuity, multivariable calculus explores partial derivatives, multiple integrals, and topics such as gradient, divergence, and curl which have critical applications in physics and engineering.

The exercise given illustrates the complexities that arise when dealing with functions of multiple variables. The concept of a limit becomes richer and more challenging to ascertain, as we must consider the fact that there are infinitely many paths along which a point can approach a limit, making evaluation and conceptual understanding more demanding but also more intriguing.