Problem 88 - Calculus 3

Question

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}} $$ a. Along the \(x\) -axis \((y=0)\) b. Along the \(y\) -axis \((x=0)\) c. Along the path \(y=x^{2}\)

Step-by-Step Solution

Answer

The limit along the x-axis and y-axis is 0, but along y=x², it's 1/2. The limit does not exist overall.

Step 1: Evaluation along the x-axis (y=0)

To evaluate the limit along the \(x\)-axis, substitute \(y=0\) into the function: \[\frac{x^2 imes 0}{x^4 + 0^2} = \frac{0}{x^4} = 0.\]Thus, as \((x, y) \to (0, 0)\) along the \(x\)-axis, the limit is 0.

Step 2: Evaluation along the y-axis (x=0)

To evaluate the limit along the \(y\)-axis, substitute \(x=0\) into the function: \[\frac{0^2 \times y}{0^4 + y^2} = \frac{0}{y^2} = 0.\]Thus, as \((x, y) \to (0, 0)\) along the \(y\)-axis, the limit is 0.

Step 3: Evaluation along the path y=x^2

To evaluate the limit along the path \(y=x^2\), substitute \(y=x^2\) into the function: \[\frac{x^2 \times x^2}{x^4 + (x^2)^2} = \frac{x^4}{x^4 + x^4} = \frac{x^4}{2x^4} = \frac{1}{2}.\]Thus, as \((x, y) \to (0, 0)\) along the path \(y = x^2\), the limit is \(\frac{1}{2}\).

Key Concepts

Understanding Path Evaluation in Multivariable Limits

In the world of multivariable calculus, path evaluation is a crucial technique used to determine the behavior of functions as they approach a specific point from different directions. Unlike single-variable calculus, where you consider a limit approaching from the left and the right, in multivariable scenarios you often have multiple paths leading to the same point.
This concept becomes vital when evaluating the limit of a function with two or more variables. The goal is to establish whether a limit is the same regardless of which path you take to approach a point. If the function consistently leads to the same value along different paths, it suggests that the limit exists. However, if a different value is reached via different paths, then the limit does not exist.

• Attempt various approaches to probe the function's continuity near the desired point.
• Ensure comprehensive evaluation by choosing strategic paths such as axes and curves.

This approach not only assists in consistency checks but also in revealing anomalies where limits might differ.

X-Axis Limit Simplified

To evaluate the x-axis limit of a multivariable function, you set the variable perpendicular to the x-axis to zero, effectively reducing the equation to one involving only the x variable. In this exercise, substituting \(y = 0\) into the original function simplifies it to \(\frac{x^{2} \times 0}{x^{4} + 0^2} = 0\).
This simplification reveals that the limit along the x-axis as \((x, y)\) approaches \((0, 0)\) is 0. This step underscores that as you move along the x-axis towards the origin, the function's value settles at zero, suggesting a consistent behavior along this path.
Understanding x-axis limits:

• Substitute the perpendicular coordinate's value to zero to isolate the variable of interest.
• Assess the resulting single-variable function for insights on limit behavior.

By doing so, you verify the function's readiness to achieve a consistent limit in this direction.

Demystifying the Y-Axis Limit

The y-axis limit follows a similar approach to the x-axis limit but instead fixes the x-axis component to zero, leading to an assessment of how the function behaves as it approaches via the y-direction. By substituting \(x = 0\) into the function, you simplify it to \(\frac{0^{2} \times y}{0^{4} + y^2} = 0\).
This indicates that as \((x, y)\) draws near \((0, 0)\) along the y-axis, the limit remains consistently zero. Just like the x-axis evaluation, it reaffirms a steady approach to zero.

• Freeze one variable by setting it to zero, in this context, \(x\).
• Evaluate the single-variable expression for convergence as it mimics y's movement to zero.

It's an essential tactic to ensure that path-based evaluations are thorough, following through all principal axes.

When a Limit Does Not Exist

A limit does not exist when different paths towards a point yield different results. In this context, after examining axis-based paths, the exploration of another non-linear path, such as \(y = x^2\), becomes crucial. By substituting \(y = x^2\) into our original equation, we derive \(\frac{x^4}{x^4 + x^4} = \frac{1}{2}\).
Since the calculated limit of \(\frac{1}{2}\) differs from the zero value obtained from both axes, it signifies a limitation inherent in multivariable limits.
Key insights when limits do not exist:

• Variable values diverge across different paths, indicating inconsistencies.
• Verify with multiple paths, including non-linear trajectories, to authenticate results.

Different outcomes from distinct paths conclusively mean the limit doesn’t exist, showcasing the diverse behavior of functions in multivariable calculus.