Problem 81
Question
For the following exercises, use algebraic techniques to evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-4 y^{4}}{x^{2}+2 y^{2}} $$
Step-by-Step Solution
Verified Answer
The limit is 0.
1Step 1: Assess Direct Substitution
Attempt to plug in \((x, y) = (0, 0)\) directly into the expression:\[\frac{x^4 - 4y^4}{x^2 + 2y^2}\]Substituting gives \(\frac{0^4 - 4(0)^4}{0^2 + 2(0)^2} = \frac{0}{0}\), indicating an indeterminate form.
2Step 2: Consider Simplifying the Expression
Recognize that the expression \( \frac{x^4 - 4y^4}{x^2 + 2y^2} \) might be simplified. Here, try to factorize the terms. Notice that \( x^4 - 4y^4 \) can be rewritten as a difference of squares:\[x^4 - 4y^4 = (x^2)^2 - (2y^2)^2 = (x^2 - 2y^2)(x^2 + 2y^2)\]This reveals a common factor with the denominator, \(x^2 + 2y^2\).
3Step 3: Simplify by Canceling Out Common Factors
Given the factorization,\[\frac{(x^2 - 2y^2)(x^2 + 2y^2)}{x^2 + 2y^2}\]we can cancel the common factor \((x^2 + 2y^2)\), provided it does not equal zero. As \((x, y)\) approaches the origin, this cancellation is valid as long as \((x^2 + 2y^2) eq 0\), at least around the limit point.This simplifies to \(x^2 - 2y^2\).
4Step 4: Evaluate the Limit of the Simplified Expression
Now evaluate the limit of the simplified expression:\[\lim _{(x, y)
ightarrow(0, 0)} (x^2 - 2y^2)\]Since this expression clearly evaluates to \(0^2 - 2(0)^2 = 0\), the limit is simply \(0\).
Key Concepts
Indeterminate FormsAlgebraic TechniquesFactorizationDifference of Squares
Indeterminate Forms
When dealing with multivariable calculus, you often encounter expressions that have an "indeterminate form." An indeterminate form is a situation where direct substitution into a limit gives a result that is not immediately clear or defined, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
These forms suggest that further algebraic manipulation is necessary to determine the limit, as they do not provide useful information by themselves.
In our exercise, substituting \((x, y) = (0, 0)\) directly into the expression \( \frac{x^4 - 4y^4}{x^2 + 2y^2} \) results in the indeterminate form \( \frac{0}{0} \). This prompts us to explore alternative ways to simplify and evaluate the expression.
These forms suggest that further algebraic manipulation is necessary to determine the limit, as they do not provide useful information by themselves.
In our exercise, substituting \((x, y) = (0, 0)\) directly into the expression \( \frac{x^4 - 4y^4}{x^2 + 2y^2} \) results in the indeterminate form \( \frac{0}{0} \). This prompts us to explore alternative ways to simplify and evaluate the expression.
Algebraic Techniques
Algebraic techniques are essential tools for simplifying complex expressions, especially in calculus when evaluating limits.
By applying these techniques, we can transform expressions into simpler forms that are easier to handle.
The key steps involve identifying possible ways to manipulate the terms of the expression, such as combining like terms, factoring, or performing polynomial long division. Each technique serves to make the expression more tractable.
In the given problem, the algebraic technique used is factorization of the numerator \(x^4 - 4y^4\), in an attempt to cancel out a common factor with the denominator \(x^2 + 2y^2\). This greatly simplifies the task of evaluating the limit.
By applying these techniques, we can transform expressions into simpler forms that are easier to handle.
The key steps involve identifying possible ways to manipulate the terms of the expression, such as combining like terms, factoring, or performing polynomial long division. Each technique serves to make the expression more tractable.
In the given problem, the algebraic technique used is factorization of the numerator \(x^4 - 4y^4\), in an attempt to cancel out a common factor with the denominator \(x^2 + 2y^2\). This greatly simplifies the task of evaluating the limit.
Factorization
Factorization is a powerful technique used to simplify expressions by breaking them down into products of simpler factors.
Understanding how to factor polynomials and recognize common patterns is essential for solving calculus problems related to limits.
In our problem, the numerator \(x^4 - 4y^4\) was factorized using the difference of squares formula:
Understanding how to factor polynomials and recognize common patterns is essential for solving calculus problems related to limits.
In our problem, the numerator \(x^4 - 4y^4\) was factorized using the difference of squares formula:
- First, recognize \(x^4\) as \((x^2)^2\) and \(4y^4\) as \((2y^2)^2\).
- This forms \((x^2)^2 - (2y^2)^2\), a difference of squares structure.
- Apply the formula \(a^2 - b^2 = (a-b)(a+b)\), resulting in the factors \((x^2 - 2y^2)(x^2 + 2y^2)\).
Difference of Squares
The difference of squares is a special polynomial expression of the form \( a^2 - b^2 \), and it can be expressed as \((a - b)(a + b)\).
This method is a cornerstone of simplifying algebraic expressions.
In our exercise, the expression \( x^4 - 4y^4 \) is identified as a difference of squares:
This method is a cornerstone of simplifying algebraic expressions.
In our exercise, the expression \( x^4 - 4y^4 \) is identified as a difference of squares:
- Recognize it as \((x^2)^2 - (2y^2)^2\).
- By using the formula, it transforms into two simpler factors \((x^2 - 2y^2)(x^2 + 2y^2)\).
- This enables the cancellation of the common factor \((x^2 + 2y^2)\) with the same term in the denominator.
Other exercises in this chapter
Problem 77
For the following exercises, evaluate the limits at the indicated values of \(x\) and \(y\) . If the limit does not exist, state this and explain why the limit
View solution Problem 80
For the following exercises, use algebraic techniques to evaluate the limit. $$ \lim _{(x, y) \rightarrow(2,1)} \frac{x-y-1}{\sqrt{x-y}-1} $$
View solution Problem 82
For the following exercises, use algebraic techniques to evaluate the limit. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3}-y^{3}}{x-y} $$
View solution Problem 84
For the following exercises, evaluate the limits of the functions of three variables. $$ \lim _{(x, y, z) \rightarrow(1,2,3)} \frac{x z^{2}-y^{2} z}{x y z-1} $$
View solution