Problem 24 - Calculus Early Transcendentals

Question

\(23-24\) Use a computer graph of the function to explain why the limit does not exist. $$\lim _{(x, y \rightarrow(0,0)} \frac{x y^{3}}{x^{2}+y^{6}}$$

Step-by-Step Solution

Answer

The limit does not exist because the function is path-dependent.

Step 1: Understand the Problem

We need to find whether the limit \( \lim _{(x, y) \rightarrow(0,0)} \frac{x y^{3}}{x^{2}+y^{6}} \) exists. This involves checking if the value the function approaches is the same from all paths leading to \((0,0)\).

Step 2: Choose Common Paths

We will evaluate the limit along two common paths: \(y = 0\) and \(x = 0\). On the path \(y = 0\), the expression simplifies to \(\frac{0}{x^2} = 0\). And on the path \(x = 0\), it simplifies to \(\frac{0}{y^6} = 0\). In both cases, the limit evaluates to 0.

Step 3: Test a Nontrivial Path

Let us test another path such as \(y = x^{1/2}\). On this path, substitute \(y\) with \(x^{1/2}\) in the function, which leads to \(\frac{x(x^{1/2})^3}{x^2 + (x^{1/2})^6} = \frac{x^{5/2}}{x^2 + x^3}\). Simplifying gives \(\frac{x^{5/2}}{x^2(1+x)}\), which tends to 0 as \(x \to 0\).

Step 4: Check Another Nontrivial Path

Try the path \(y = x^2\) which leads to \(\frac{x(x^2)^3}{x^2 + (x^2)^6} = \frac{x^7}{x^2+x^{12}} \). Simplifying, it becomes \(\frac{x^7}{x^2(1+x^{10})} = \frac{x^5}{1+x^{10}}\). Here, we can see that as \(x \to 0\), it evaluates to 0.

Step 5: Analyze with Graphing Software (Conceptual)

Use a graphing software to plot the function in 3D. You'll observe that as \((x,y)\) approaches \((0,0)\), the function value does not stabilize to a single value, despite individual paths appearing to approach 0. This suggests the behavior is path-dependent and different along linear and non-linear paths.

Step 6: Conclude the Non-existence

The function exhibits different approaches (or does not approach smoothly) via different nontrivial paths observed through graphing. Therefore, the behavior suggests path dependency and indicates the limit does not exist.

Key Concepts

Multivariable Calculus

Multivariable Calculus is an extension of calculus into higher dimensions. It deals with functions that have more than one variable, such as \( f(x, y) \). This allows us to study surfaces and curves in space, not just lines.
In problems involving limits, we often try to find how a function behaves as it approaches a point from different directions.
Working with multiple variables adds complexity, as there are infinitely many paths to approach any given point. This property makes understanding limits in multivariable functions quite fascinating and sometimes tricky.

Path Dependency

Path dependency in calculus refers to whether the value a function approaches as it nears a point depends on the path taken.
In single-variable calculus, we move along a straight line towards a point. However, in multivariable calculus, we can approach from any direction in a plane.
Path dependency becomes crucial because if a limit gives different results based on the path chosen, then the limit does not exist.

• Linear paths are simple direct lines.
• Non-linear paths can curve and bend, providing additional insight.

Understanding path dependency helps recognize why some limits do not exist if the value isn't consistent across all paths.

Limit Analysis

Limit Analysis refers to the process of determining if a function approaches a particular value as its variables approach certain points.
We often choose multiple paths to evaluate the limit to identify if the function consistently approaches the same value.
A typical strategy involves choosing both straightforward paths and more complex ones, such as:

• The line \( y = 0 \) which evaluates the function without altering the x-coordinate.
• A curved path like \( y = x^{1/2} \) to explore different behaviors.

Analyzing limits through different paths reveals the function's consistency and helps predict the behavior near the point of interest.

Non-existence of Limits

The Non-existence of Limits occurs when a function does not approach a single, well-defined value as its variables approach a point.
In multivariable calculus, this often happens when different paths yield different results.
For example, a function \( f(x, y) \) might appear to converge along some paths but behave erratically along others. If even one path leads to a different result, the overall limit can't exist.

• Graphs of functions can visually display this inconsistency.
• Testing through software can highlight the path dependence visually.

Recognizing non-existence is crucial for understanding the complex behavior of multivariable functions.