Problem 104 - Calculus Volume 3

Question

[T] Evaluate \(\lim _{(x, y) \rightarrow(0,0)} \frac{-x y^{2}}{x^{2}+y^{4}}\) by plotting the function using a CAS. Determine analytically the limit along the path \(x=y^{2}\).

Step-by-Step Solution

Answer

The limit along the path \(x = y^2\) is \(\frac{-1}{2}\).

Step 1: Understanding the Limit Problem

We are tasked with finding the limit of the function \( f(x, y) = \frac{-xy^2}{x^2 + y^4} \) as \((x, y)\) approaches \((0, 0)\). To do this, we will first approach the limit along the path \(x = y^2\), meaning we will substitute \(x\) with \(y^2\) to simplify our analysis.

Step 2: Substitute Path into the Function

Substitute \(x = y^2\) into the function to see how it behaves along this path: \[f(x, y) = \frac{-(y^2)y^2}{(y^2)^2 + y^4} = \frac{-y^4}{y^4 + y^4} = \frac{-y^4}{2y^4} = \frac{-1}{2}.\]Now, simplify the expression to find the limit along this path.

Step 3: Evaluate the Simplified Expression

After substituting and simplifying, the expression reduces to a constant value: \(\frac{-1}{2}\). This indicates that along the path \(x = y^2\), the function does not depend on \(y\) for the limit as \((x, y)\) approaches \((0, 0)\).

Step 4: Determine the General Behavior

To confirm the limit exists and is the same from all directions, additional considerations or plots might be needed using a computer algebra system (CAS). However, as directed by the problem, we have analytically determined the limit along the specified path.

Key Concepts

Limits of Functions

Understanding limits is fundamental in multivariable calculus. A limit describes the value that a function approaches as the input gets closer to some point. For functions of two variables, such as our exercise function \( f(x, y) = \frac{-xy^2}{x^2 + y^4} \), the limit as \((x,y)\) approaches a point can be more complex than single-variable cases.

Limits involving two variables require considering all possible directions of approach to the point. Unlike single-variable calculus where direction is linear (from right and left), multivariable calculus involves directions from all angles on the plane. This makes them intriguing yet challenging.

• Evaluate limits by analyzing paths.
• Confirm that a function approaches the same value from different directions to support the existence of a limit.

When a function approaches the same limit from every direction, the limit exists. Otherwise, it might depend on the path taken or not exist at all.

Approaching a Point

Approaching a point in multivariable calculus involves observing the behavior of a function as the variables get arbitrarily close to a specific point. In our case, that point is \((0, 0)\).

To approach \((0, 0)\), one can analyze several possible paths, such as lines, curves, or other algebraic relationships between \(x\) and \(y\). Each path provides insight into how the function behaves.

In our specific problem, we analyzed the path where \(x = y^2\).

• This approach simplifies the function, allowing us to examine the behavior more clearly along that path.
• If evaluated limits along multiple paths yield the same result, we gain confidence that this limit is consistent.

Being methodical about approaching the point helps determine the true behavior of the function near the point of interest.

Path Analysis

Path analysis involves studying how a function behaves along specific paths to evaluate its limits. It highlights the importance of choosing appropriate paths that simplify the analysis.

In the given problem, taking the path \(x = y^2\) offered a straightforward way to simplify the function, which helped in evaluating the limit effortlessly.

• Substitute the chosen path into the function.
• Analyze the resulting expression to determine the limit.
• Repeat with other paths to verify if the function consistently approaches the same limit.

Ultimately, path analysis provides a means to explore the potentially problematic nature of multivariable limits by evaluating multiple scenarios.

Computer Algebra System

A Computer Algebra System (CAS) can be a handy tool when working with complex calculations, such as evaluating multivariable limits. CAS software can handle the algebraic manipulations and graphing necessary to explore the behavior of functions in different regions of their domain.

For the original exercise, plotting the function \( f(x, y) = \frac{-xy^2}{x^2 + y^4} \) using a CAS helps visualize the behavior as \((x, y)\) approaches \((0, 0)\).

• A CAS can provide a visual confirmation of analytical results.
• Graphical outputs can reveal unusual behavior or confirm expectations about limits.

This visualization and computational strength ensures comprehensive understanding, aiding both learning and exploration of complex multivariable functions.