/*! 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 1 Find the limits, if they exist, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 1

Find the limits, if they exist, or type \(D N E\) for any which do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{1 x^{2}}{3 x^{2}+4 y^{2}}$$ 1) Along the \(x\) -axis: ___________. 2) Along the \(y\) -axis: ___________. 3) Along the line \(y=m x\) : ___________. 4) The limit is: ___________.

Short Answer

Expert verified
1) Along the \(x\)-axis: \(1/3\). 2) Along the \(y\)-axis: \(0\). 3) Along the line \(y=mx\) : \(\frac{1}{3+4m^2}\). 4) The limit is: D N E.

Step by step solution

01

Evaluate along the x-axis (y = 0)

When y = 0, the function becomes: \(\lim_{x\rightarrow 0}\frac{1x^2}{3x^2}\) Notice that the y term is eliminated, so only the x terms are remaining. Divide both the numerator and denominator by x^2 to simplify: \(\lim_{x\rightarrow 0}\frac{1}{3}\) Since there are no remaining x terms, this limit is a constant, and its value is \(1/3\).
02

Evaluate along the y-axis (x = 0)

When x = 0, the function becomes: \(\lim_{y\rightarrow 0}\frac{1(0)^2}{3(0)^2 + 4y^2}\) Notice that the x term is eliminated, so only the y terms are remaining. This simplifies to: \(\lim_{y\rightarrow 0}(0)\) As y approaches 0, this limit equates to zero.
03

Evaluate along the line y = mx

Now, we will consider any other line given by y = mx. Substitute y = mx into the function: \(\lim_{(x, mx)\rightarrow (0,0)}\frac{1x^{2}}{3x^{2}+4(mx)^{2}}\), which simplifies to: \(\lim_{x\rightarrow 0}\frac{1x^{2}}{3x^{2}+4m^2x^2}\) Now, factor out x^2 from both the numerator and denominator: \(\lim_{x\rightarrow 0}\frac{x^2}{x^2(3+4m^2)}\) Eliminate x^2 terms from both the numerator and denominator: \(\lim_{x\rightarrow 0}\frac{1}{3+4m^2}\) As x approaches 0, this limit becomes a constant value based on the slope m. It is neither zero nor \(1/3\), and it depends on the value of m.
04

Determine the limit

In Steps 1, 2, and 3, we found the limits of the given function along the x-axis, y-axis, and the line y = mx. We found that the limit is inconsistent as it varies along different lines. Therefore, the limit of the function as (x, y) approaches (0,0) does not exist. 4) The limit is: D N E (Does Not Exist).

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!

Key Concepts

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

Limit Along a Path
In multivariable calculus, understanding limits along a path is crucial. When we say a function has a limit as it approaches a point, we're imagining getting closer from many routes or paths. However, each path might yield a different result. That's what makes checking the limit along specific paths important.

In the given exercise, we investigate limits along specific paths:
  • Along the x-axis where y = 0
  • Along the y-axis where x = 0
  • Along a line y = mx, where m is a constant
Each of these paths simplifies the problem uniquely as we approach the origin (0,0). By examining the different results obtained for each path, we can determine whether a consistent limit is achieved as we approach the point from various directions.
Non-Existence of Limits
Limits may not always exist in multivariable calculus, particularly if they yield different results depending on the path taken. In the exercise, when evaluating each path, different results indicate the limit doesn't exist at the point.

For our function, the limit along the x-axis is 1/3, while along the y-axis it is 0. Additionally, as we approach using a general line with slope m, the result varies with m. Since the values differ with each direction, a single limit value cannot be established.

This scenario leads us to conclude that the limit at the point in question does not exist. Such conclusions are important because they highlight the nature of multivariable functions, which can behave unpredictably as we approach specific points.
Multivariable Calculus
Multivariable calculus expands calculus concepts into more dimensions, making it possible to analyze functions with several variables. Limits play a substantial role in understanding the behavior of these functions. Just as single-variable limits help in examining how functions act near specific points on a line, multivariable limits serve a similar purpose in planes or higher-dimensional spaces.

In this context, the challenge lies in exploring multiple paths. This involves looking at planes, lines, or curves through our point of interest.

In educational contexts, mastering this concept helps build up to more complex ideas like gradients and surface integrals. We ground our intuition for changes in functions like slopes or curvatures鈥攁 fundamental part of advanced calculus studies.
Path Dependency in Limits
Path dependency refers to how a limit's outcome can change based on the path taken towards that point. In other words, it's how specific directions affect results as you try to approach a single point in a multivariable setting. For this exercise, learning about paths helps demonstrate why limits may differ. A function might yield a consistent result for certain paths while varying drastically with others.

With the function given in our limit problem, it highlights path dependency clearly:
  • The limit along the x-axis results in 1/3
  • The limit along the y-axis results in 0
  • The limit varies when moving along a line with slope m
Path dependency underscores complexity in higher dimensions and reflects real-world phenomena where direction changes outcomes significantly.

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

In a simple electric circuit, Ohm's law states that \(V=I R,\) where \(\mathrm{V}\) is the voltage in volts, I is the current in amperes, and \(\mathrm{R}\) is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.03 volts per second and, as the resistor heats up, the resistance is increasing at 0.02 ohms per second. When the resistance is 100 ohms and the current is 0.02 amperes, at what rate is the current changing? ___________ amperes per second .

Design a rectangular milk carton box of width \(w\), length \(l\), and height \(h\) which holds \(500 \mathrm{~cm}^{3}\) of milk. The sides of the box cost 1 cent \(/ \mathrm{cm}^{2}\) and the top and bottom cost \(2 \mathrm{cent} / \mathrm{cm}^{2}\). Find the dimensions of the box that minimize the total cost of materials used. dimensions = __________.

Find all directions in which the directional derivative of \(f(x, y)=y e^{-x y}\) is 1 at the point (0,2) .

The Heat Index, \(I,\) (measured in apparent degrees \(F\) ) is a function of the actual temperature \(T\) outside (in degrees \(\mathrm{F}\) ) and the relative humidity \(H\) (measured as a percentage). A portion of the table which gives values for this function, \(I(T, H),\) is reproduced in Table 10.3 .11 . $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\ \hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ a. State the limit definition of the value \(I_{T T}(94,75)\). Then, estimate \(I_{T T}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including units. b. State the limit definition of the value \(I_{H H}(94,75) .\) Then, estimate \(I_{H H}(94,75),\) and write one complete sentence that carefully explains the meaning of this value, including units. c. Finally, do likewise to estimate \(I_{H T}(94,75),\) and write a sentence to explain the meaning of the value you found.

The concentration of salt in a fluid at \((x, y, z)\) is given by \(F(x, y, z)=\) \(2 x^{2}+3 y^{4}+2 x^{2} z^{2} \mathrm{mg} / \mathrm{cm}^{3}\). You are at the point (-1,1,-1) . (a) In which direction should you move if you want the concentration to increase the fastest? direction: _________. (Give your answer as a vector.) (b) You start to move in the direction you found in part (a) at a speed of \(5 \mathrm{~cm} / \mathrm{sec} .\) How fast is the concentration changing? rate of change \(=\) ______________.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.