Problem 82 - Calculus 3

Question

For the following exercises, use algebraic techniques to evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x-y} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Problem

We're asked to find the limit of the function \( \frac{x^{3} - y^{3}}{x - y} \) as \( (x, y) \) approaches \( (0, 0) \). This involves determining how the function behaves as both \( x \) and \( y \) get closer and closer to zero.

Step 2: Applying Algebraic Techniques

Since the expression is a form that suggests indeterminacy \( \frac{0}{0} \), we'll use algebraic manipulation. The difference of cubes formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \) can be applied here.

Step 3: Factoring the Numerator

Apply the formula \( x^3 - y^3 = (x-y)(x^2 + xy + y^2) \) to the numerator of the function. Now, the expression becomes:\[ \frac{(x-y)(x^2 + xy + y^2)}{x-y} \]

Step 4: Canceling Common Factors

Notice that \( x-y \) in the numerator and denominator are common factors, so they can be cancelled out (assuming \( x eq y \) to avoid division by zero). Therefore, the expression simplifies to:\[ x^2 + xy + y^2 \]

Step 5: Evaluating the Limit

Now, we evaluate the limit for the simplified expression \( x^2 + xy + y^2 \) as \( (x, y) \) approaches \( (0, 0) \):\[ \lim_{(x, y) \to (0, 0)} (x^2 + xy + y^2) = 0^2 + 0\cdot0 + 0^2 = 0 \]

Step 6: Conclude the Solution

The limit of the original expression as \((x, y)\) approaches \((0, 0)\) is 0, after algebraic simplification and evaluation.

Key Concepts

Algebraic Techniques

In studying multivariable calculus, particularly when evaluating limits, algebraic techniques play a crucial role. These techniques help simplify expressions, especially when they initially appear to be indeterminate. Indeterminate forms, such as \( \frac{0}{0} \), require careful manipulation to evaluate appropriately.
In this exercise, the expression \( \frac{x^{3} - y^{3}}{x - y} \) initially presents as an indeterminate form when \( (x, y) \) approaches \( (0, 0) \). Algebraic techniques allow us to reconfigure the expression to bypass this indeterminacy.
This involves applying specific algebraic forms like difference of cubes which enables further simplification, making it possible to integrate further limits or analytic strategies. Recognizing that algebraic manipulation can transform a complex problem into a straightforward one is a key skill in calculus.

Limit Evaluation

Limit evaluation is the process of determining the behavior of a function as the inputs approach a certain point. In multivariable calculus, assessing the limit of a function as the variables converge to a given point requires an understanding of how the function simplifies over different conditions.
For the exercise \( \lim_{(x, y) \to (0,0)} \frac{x^{3}-y^{3}}{x-y} \), once we apply the algebraic techniques, it becomes possible to directly evaluate the simplified expression. Removing the factor of \( x-y \) in both numerator and denominator achieves this simplification.
After cancelling the common factor, the limit becomes the straightforward evaluation of \( x^2 + xy + y^2 \) at \( (0, 0) \), which leads directly to the result 0. This step demonstrates the importance of limit evaluation for navigating through complex functions and extracting meaningful results.

Difference of Cubes Formula

The difference of cubes formula is an essential algebraic identity that is especially useful when dealing with polynomial expressions like \( x^3 - y^3 \). The formula states:\[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]This identity is leveraged to simplify expressions by factoring, as seen in this exercise.
Applying the formula to \( x^3 - y^3 \) translates it into \((x-y)(x^2 + xy + y^2)\). This directly helps to cancel the common factor of \( x-y \) in our limit expression, leading to a simpler function that can be evaluated easily as the limit approaches \( (0, 0) \).
Understanding how to apply the difference of cubes formula can simplify numerous algebraic challenges in calculus, especially when managing expressions involving three-dimensional variables approaching specific points.