/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Evaluate the following limits. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 55

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}}$$

Short Answer

Expert verified
Question: Evaluate the limit as (x, y, z) approach (1, 1, 1) for the function $$\frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}}$$. Answer: The limit of the given function as (x, y, z) approach (1, 1, 1) is 0.

Step by step solution

01

Identify the limit point

The point that (x, y, z) are approaching is (1, 1, 1).
02

Factor the numerator and denominator

We want to simplify the given expression, so first, we factor the numerator: $$yz - xy - xz - x^{2} = -(x^2 - xy - xz + yz)$$ We can observe that x is a common factor, so we factor it : $$ -(x(x - y - z + yz))$$ Now, factor the denominator: $$yz + xy + xz - y^2 = yz - y^2 + xy + xz$$ Notice that y is a common factor, so we factor it: $$y(yz - y + x(x + z))$$ So now we have: $$\frac{-(x(x - y - z + yz))}{y(yz - y + x(x + z))}$$
03

Evaluate the limit

Now, we can evaluate the limit as (x, y, z) approach (1, 1, 1): $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{-(1(1 - 1 - 1 + 1))}{1(1 - 1 + 1(1 + 1))}$$ After plugging in the values 1 for x, y, and z, we get: $$\frac{-(1(0))}{1(1)} = \frac{0}{1}$$ So the final answer is: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}} = \boxed{0}$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Identify and briefly describe the surfaces defined by the following equations. $$y^{2}-z^{2}=2$$

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)\) The data may be plotted as a scatterplot in the \(x y\) -plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Let the equation of the best-fit line be \(y=m x+b,\) where the slope \(m\) and the \(y\) -intercept \(b\) must be determined using the least squares condition. First assume that there are three data points \((1,2),(3,5),\) and \((4,6) .\) Show that the function of \(m\) and \(b\) that gives the sum of the squares of the vertical distances between the line and the three data points is $$ \begin{aligned} E(m, b)=&((m+b)-2)^{2}+((3 m+b)-5)^{2} \\ &+((4 m+b)-6)^{2} \end{aligned}. $$ Find the critical points of \(E\) and find the values of \(m\) and \(b\) that minimize \(E\). Graph the three data points and the best-fit line.

Find the points (if they exist) at which the following planes and curves intersect. $$y=2 x+1 ; \quad \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$

An important derivative operation in many applications is called the Laplacian; in Cartesian coordinates, for \(z=f(x, y),\) the Laplacian is \(z_{x x}+z_{y y} .\) Determine the Laplacian in polar coordinates using the following steps. a. Begin with \(z=g(r, \theta)\) and write \(z_{x}\) and \(z_{y}\) in terms of polar coordinates (see Exercise 64). b. Use the Chain Rule to find \(z_{x x}=\frac{\partial}{\partial x}\left(z_{x}\right) .\) There should be two major terms, which, when expanded and simplified, result in five terms. c. Use the Chain Rule to find \(z_{y y}=\frac{d}{\partial y}\left(z_{y}\right) .\) There should be two major terms, which, when expanded and simplified, result in five terms. d. Combine parts (b) and (c) to show that $$z_{x x}+z_{y y}=z_{r r}+\frac{1}{r} z_{r}+\frac{1}{r^{2}} z_{\theta \theta}$$

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=x y \cos x y$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.