Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Limit Type

The given limit is \( \lim_{(x, y) \to (0,0)} \frac{x^{4} - y^{4}}{x^{2} + y^{2}} \). This is a multivariable limit since it involves two variables: \(x\) and \(y\).

Step 2: Substitute Directly to Check Undefined Form

Direct substitution of \(x=0\) and \(y=0\) into the given function results in the form \(\frac{0 - 0}{0 + 0} = \frac{0}{0}\), which is undefined. This suggests that further investigation is needed to evaluate the limit.

Step 3: Examine Using Polar Coordinates

Convert to polar coordinates where \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). The expression becomes \(\frac{(r^4)(\cos^4(\theta) - \sin^4(\theta))}{r^2}= r^2(\cos^4(\theta) - \sin^4(\theta))\)

Step 4: Evaluate the Modified Limit

The expression reduces to \(r^2(\cos^4(\theta) - \sin^4(\theta))\). As \(r \to 0\), regardless of \(\theta\), the expression approaches zero \((r^2 \cdot \, \text{some bounded function}) \to 0\cdot\text{constant} = 0\).

Step 5: Use Path Approach for Verification

Check by approaching \( (0,0) \) along different paths. For example, using the path \(y=x\), the expression simplifies to \(\frac{x^4 - x^4}{x^2 + x^2} = 0\); using \(y=0\), it simplifies to \(\frac{x^4}{x^2} = x^2\) which goes to 0 as \(x\to 0\). All tested paths yield the same result, confirming the limit.

Key Concepts

Polar Coordinates in Limits

When evaluating multivariable limits, such as the problem involving \( \lim_{(x, y) \to (0,0)} \frac{x^{4} - y^{4}}{x^{2} + y^{2}} \), polar coordinates can be a powerful tool. This approach transforms \(x\) and \(y\) into terms of \(r\) and \(\theta\), using the substitutions \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).
This substitution can simplify the evaluation of limits where direct substitution yields an undefined form.

• With our example, substituting polar expressions, the original function becomes \(\frac{(r^4)(\cos^4(\theta) - \sin^4(\theta))}{r^2}= r^2(\cos^4(\theta) - \sin^4(\theta))\).
• As \(r \rightarrow 0\), \(r^2\) also approaches 0, effectively governing the limit's behavior to make it approach zero for any \(\theta\).
  

This computation shows how polar coordinates can often simplify such limits into more manageable expressions, ultimately leading to easier evaluation of whether a limit exists or not.

Undefined Forms in Calculus

An expression that results in \(\frac{0}{0}\) is known as an undefined form, a common scenario in calculus limit problems.
These forms suggest that simple substitutions aren't enough to find a limit, urging the need for alternative strategies.

• In this particular limit exercise, substituting \(x=0\) and \(y=0\) results in \(\frac{0 - 0}{0 + 0} = \frac{0}{0}\).
• This signals that further analysis is necessary because the result is undefined, meaning the limit could exist, not exist, or equal different values depending on the path taken.
  

To resolve these situations, mathematicians often employ strategies like polar coordinate transformations or evaluating the function along different paths, avoiding the direct approach which leads to indeterminacy. Recognizing undefined forms hints at the richness of a function's behavior near points of interest, necessitating more sophisticated techniques for conclusive results.

Path Approach in Calculus

The path approach is a method to evaluate limits by examining the function along various trajectories that lead to the same point. This approach helps determine if a multivariable limit exists or varies by path.
By assessing different specific paths to approach \((0,0)\), additional insights into the function are obtained:

• For the given scenario, one such path is \(y=x\), leading to \(\frac{x^4 - x^4}{x^2 + x^2} = 0\).
• Another path is \(y=0\), which simplifies to \(\frac{x^4}{x^2} = x^2\), also reaching 0 as \(x\to 0\).

All these tested paths give the same limit, illustrating that the function behaves consistently regardless of the approach. Such consistency confirms that the limit exists and is path-independent, an essential check for verifying multivariable limits where direct calculation is inconclusive. This method ensures that no other distinct outcomes arise from other untested paths, guaranteeing the robustness of the solution.