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Chapter 13: Problem 24

Find the limits by rewriting the fractions first. $$\lim _{(x, y) \rightarrow(2,2)} \frac{x-y}{x^{4}-y^{4}}$$

Short Answer

Expert verified
The limit is \(\frac{1}{32}\).

Step by step solution

01

Understand the Problem

We are asked to find the limit of the function \( \frac{x-y}{x^4-y^4} \) as \((x, y) \to (2, 2)\). This involves analyzing the behavior of the function when both \(x\) and \(y\) approach 2 simultaneously.
02

Factor the Denominator

The denominator \(x^4 - y^4\) is a difference of two powers and can be factored as \((x^2 + y^2)(x+y)(x-y)\). It is helpful to factor to possibly cancel terms in the numerator.
03

Cancel Common Factors

Rewrite the original fraction: \(\frac{x-y}{(x-y)(x^2 + y^2)(x+y)}\). Cancel the \((x-y)\) in the numerator and the denominator: \(\frac{1}{(x^2 + y^2)(x+y)}\).
04

Substitute Limit Values

With the simplification, substitute \(x = 2\) and \(y = 2\) into the simplified expression: \(\frac{1}{((2)^2 + (2)^2)(2+2)}\).
05

Calculate the Value

Calculate the expression: \(\frac{1}{(4+4)(4)} = \frac{1}{32}\). This is the value of the limit as \((x, y) \to (2, 2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Powers
When faced with an expression like \(x^4 - y^4\), noticing that it is a difference of powers can simplify our problem-solving process. This term specifically involves the subtraction of similar exponential terms, in this case raising both \(x\) and \(y\) to the fourth power. The difference of powers can often be factored into simpler, more manageable parts. This is particularly useful in limit evaluation problems where simplifying the expression can reveal removable discontinuities.
  • Recognizing patterns in exponents can expedite the calculation process.
  • The difference of squares is a specific case but universally applicable when factorizing higher degrees.
  • \(x^4 - y^4\) can be expressed as \((x^2 - y^2)(x^2 + y^2)\), or further as \((x-y)(x+y)(x^2 + y^2)\).
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into products of simpler polynomials. This is a crucial skill in mathematical problem-solving as it allows for greater manipulation and simplification of expressions. When evaluating limits, factoring can reveal common factors that could eliminate undefined expressions, such as a \(0/0\) indeterminate form.
  • In our case, \(x^4 - y^4\) factors into \((x-y)(x+y)(x^2+y^2)\), using the identity for the difference of powers.
  • This transformation is vital to simplifying the expression of limits.
  • The art of factoring involves recognizing and applying algebraic identities effectively.
Once factored, these components can potentially cancel with terms in the numerator, reducing the complexity of calculating the limit.
Common Factor Cancellation
In expressions involving limits, finding and canceling common factors is often a step towards simplifying an expression and resolving indeterminate forms. This was clearly depicted when \((x-y)\) was both in the numerator and denominator in the fraction \(\frac{x-y}{x^4-y^4}\). By factoring and then canceling these terms, we effectively reduce the expression to a simpler form.

Simplified Form

Once \((x-y)\) was canceled out:
  • The limit expression became \(\frac{1}{(x^2 + y^2)(x+y)}\) after cancellation.
  • Minimizing complexity through cancellation helps in touching upon the core value of the limit function.
Multivariable Limits
Multivariable limits involve evaluating the behavior of functions as two or more variables approach certain values simultaneously. These problems require comprehension of paths and continuity in higher dimensions, making them more intricate than single-variable limits.
  • In this context, both \(x\) and \(y\) approach the value of 2 at the same time.
  • By simplifying the original expression, substituting \(x = 2\) and \(y = 2\) becomes straightforward without resulting in indeterminate forms.
When evaluating these kinds of limits:
  • Always try to simplify the expression first, where possible, before substitution.
  • Ensure the path of approach is such that it satisfies the conditions for continuity and the existence of the limit.
  • The calculation leads to a concrete limit value of \(\frac{1}{32}\) in our case study.

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Most popular questions from this chapter

During the 1920 s, Charles Cobb and Paul Douglas modeled total production output \(P\) (of a firm, industry, or entire economy) as a function of labor hours involved \(x\) and capital invested \(y\) (which includes the monetary worth of all buildings and equipment). The Cobb-Douglas production function is given by $$P(x, y)=k x^{\alpha} y^{1-\alpha}$$ where \(k\) and \(\alpha\) are constants representative of a particular firm or economy. a. Show that a doubling of both labor and capital results in a doubling of production \(P\) b. Suppose a particular firm has the production function for \(k=\) 120 and \(\alpha=3 / 4 .\) Assume that each unit of labor costs $$ 250$ and each unit of capital costs $$ 400, and that the total expenses for all costs cannot exceed $$ 100,000 . Find the maximum production level for the firm.

In economics, the usefulness or utility of amounts \(x\) and \(y\) of two capital goods \(G_{1}\) and \(G_{2}\) is sometimes measured by a function \(U(x, y) .\) For example, \(G_{1}\) and \(G_{2}\) might be two chemicals a pharmaceutical company needs to have on hand and \(U(x, y)\) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If \(G_{1}\) costs \(a\) dollars per kilogram, \(G_{2}\) costs \(b\) dollars per kilogram, and the total amount allocated for the purchase of \(G_{1}\) and \(G_{2}\) together is \(c\) dollars, then the company's managers want to maximize \(U(x, y)\) given that \(a x+b y=c .\) Thus, they need to solve a typical Lagrange multiplier problem. Suppose that $$U(x, y)=x y+2 x$$ and that the equation \(a x+b y=c\) simplifies to $$2 x+y=30$$ Find the maximum value of \(U\) and the corresponding values of \(x\) and \(y\) subject to this latter constraint.

Use a CAS to plot the implicitly defined level surfaces. $$x^{2}+z^{2}=1$$

Find the linearizations \(L(x, y, z)\) of the functions at the given points. \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\) at a. (1,0,0) b. (1,1,0) c. (1,2,2)

Gives a function \(f(x, y, z)\) and a positive number \(\epsilon .\) In each exercise, show that there exists a \(\delta>0\) such that for all \((x, y, z)\) $$\sqrt{x^{2}+y^{2}+z^{2}}<\delta \Rightarrow|f(x, y, z)-f(0,0,0)|<\epsilon$$ $$f(x, y, z)=\frac{x+y+z}{x^{2}+y^{2}+z^{2}+1}, \quad \epsilon=0.015$$
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