/*! 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 6 Explain why evaluating a limit a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 6

Explain why evaluating a limit along a finite number of paths does not prove the existence of a limit of a function of several variables.

Short Answer

Expert verified
Answer: Evaluating a limit along a finite number of paths does not prove the existence of a limit for a function of several variables because there may be directions where the limit does not exist or approaches a different value. To draw any conclusion about the existence of the limit, we need to analyze the function for all possible paths, which may lead to varying results. Continuity at a point serves as a helpful tool to prove the existence of a limit.

Step by step solution

01

Introduction to Limits in Multivariable Functions

A limit in a function of several variables is similar to the concept of a limit in a single-variable function. The limit of a function f(x,y) at a point (a,b) is said to exist if the function approaches a unique value L as the inputs approach the point (a,b) from all possible paths. Mathematically, this is expressed as: \(\lim_{(x,y) \to (a,b)} f(x,y) = L\)
02

Evaluating Limits Along Different Paths

When considering the existence of a limit for a function of several variables, we must consider all possible paths to the point (a,b). Evaluating a limit along a finite number of paths (e.g., x-axis and y-axis) doesn't guarantee the limit exists because there may be directions where the limit does not exist or approaches a different value. A classic example of this is the following function: \(f(x, y) = \dfrac{xy}{x^2+y^2}\) for \((x,y) \neq (0,0)\) and \(f(0,0) = 0\). As can be seen, even if the function is evaluated through multiple paths and the result is finite, unwarranted assumptions might lead us to believe that a conclusion on the limit existence is reached.
03

The Role of Continuity

Continuity of a multivariable function at a point requires that the limit should exist at that point, and the value of the function at that point should be equal to the limit. If the function is continuous at a point, then we can be sure that the limit exists. For the given function \(f(x, y)\), if we could prove it's continuous at (0,0), we could conclude that its limit exists. However, for this particular function, it's not continuous, so we can't use that to prove the existence of the limit.
04

Conclusion

Evaluating a limit along a finite number of paths for a multivariable function does not guarantee the existence of the limit. To draw any conclusion about the existence of the limit, we need to analyze the function for all possible paths, which may lead to varying results. Continuity at a point serves as a helpful tool to prove the existence of a limit.

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

Find an equation of the plane passing through the point (3,2,1) that slices off the region in the first octant with the least volume.

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$h(x, y, z)=\sqrt[4]{z^{2}-x z+y z-x y}.$$

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$

Find the points at which the plane \(a x+b y+c z=d\) intersects the \(x-y-\), and \(z\) -axes.

Consider the following functions \(f.\) a. Is \(f\) continuous at (0,0)\(?\) b. Is \(f\) differentiable at (0,0)\(?\) c. If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\) d. Determine whether \(f_{x}\) and \(f_{y}\) are continuous at \((0,0).\) e. Explain why Theorems 5 and 6 are consistent with the results in parts \((a)-(d).\) $$f(x, y)=\sqrt{|x y|}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision 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.