Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Consider the Limit Along the x-axis

Along the x-axis, we set \( y = 0 \). The limit becomes \( \lim_{x \to 0} \frac{x^2 \cdot 0^2}{x^2 + 0^4} = \lim_{x \to 0} 0 = 0 \).

Step 2: Consider the Limit Along the y-axis

Along the y-axis, set \( x = 0 \). The expression simplifies to \( \lim_{y \to 0} \frac{0^2 \cdot y^2}{0^2 + y^4} = \lim_{y \to 0} 0 = 0 \).

Step 3: Approach Along the Line y = x

Substitute \( y = x \). The limit becomes \( \lim_{x \to 0} \frac{x^2 \, x^2}{x^2 + x^4} = \lim_{x \to 0} \frac{x^4}{x^2(1 + x^2)} = \lim_{x \to 0} \frac{x^2}{1 + x^2} = 0 \).

Step 4: Approach Along the Line y = x^2

Substitute \( y = x^2 \). The limit becomes \( \lim_{x \to 0} \frac{x^2 \, (x^2)^2}{x^2 + (x^2)^4} = \lim_{x \to 0} \frac{x^6}{x^2(1 + x^6)} = \lim_{x \to 0} \frac{x^4}{1 + x^6} = 0 \).

Step 5: Conclude the Limit

Since all paths considered (x-axis, y-axis, \( y = x \), and \( y = x^2 \)) lead to \(0\), the limit exists and is \(0\).

Key Concepts

Approaching Paths

When calculating multivariable limits, choosing various paths towards the point of interest is essential. It helps determine if the limit exists and what it might be. For the given exercise, the goal was to find the limit as \((x, y)\) approaches \((0, 0)\).
Various paths included:

• Along the x-axis, where \(y = 0\)
• Along the y-axis, where \(x = 0\)
• Along the line \(y = x\)
• Along the curve \(y = x^2\)

By substituting these into the original function, you can simplify and calculate the limit for each path. If all computations return the same value, it suggests the limit might be consistent across various routes, indicating the limit could exist.

Limit Existence

Determining whether a limit exists is crucial in multivariable calculus. In our investigation, each path we considered led to the same limit value of 0.
This consistency implies that the limit likely exists at \((0, 0)\), and is specifically zero. Remember, for a multivariable limit to exist:

• All chosen paths must approach the same limit value.
• There shouldn't be any path leading to a different result.

If even a single path differs, the limit does not exist, highlighting the role of multiple path checks in ensuring accuracy.

Coordinate Axes

The coordinate axes involve focusing on how the function behaves as you approach along the x-axis and y-axis separately.
For example, approaching along the x-axis means setting \(y = 0\) and assessing the limit of the resulting single-variable function as \(x\) approaches 0. Similarly, for the y-axis approach, set \(x = 0\). This often simplifies the evaluation process drastically:

• Less complex algebra as one variable is nullified.
• Identify discrepancies earlier due to the relatively simple calculations.

The axis approach is often the first step in checking for multivariable limit existence due to its straightforwardness.

Substitution Method

The substitution method is a powerful tool in examining multivariable limits, allowing you to explore the limit from different angles. For example:

• By substituting \(y = x\), the function morphs, and you analyze its behavior through a single-variable expression.
• Similarly, substituting \(y = x^2\) provides another perspective.

These substitutions help reveal patterns and potential limit inconsistencies. If the outcome remains stable across several substitutions, it reinforces the suspected limit value.
Bear in mind, the substitution method's strength lies in exposing variations that might not be evident from coordinate axis checks alone, thus offering a more comprehensive understanding of the function's behavior near the point of interest.