/*! 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 16 Find the limits in Exercises \(1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 16

Find the limits in Exercises \(13-24\) by rewriting the fractions first. $$\lim _{(x, y) \rightarrow(2,-4) \atop x \neq-4, x \neq x^{2}} \frac{y+4}{x^{2} y-x y+4 x^{2}-4 x}$$

Short Answer

Expert verified
The limit is undefined or requires more paths evaluation.

Step by step solution

01

Understand the Problem

The problem requires us to find the limit of a function \( \lim _{(x, y) \rightarrow(2,-4)} \frac{y+4}{x^{2} y - x y + 4 x^{2} - 4 x} \) as \( (x, y) \) approaches \( (2, -4) \). We must look at the behavior of the function as \( x \to 2 \) and \( y \to -4 \).
02

Simplify the Expression

First, observe the expression \( y + 4 \). When \( y \to -4 \), this becomes \( 0 \). Now, investigate the denominator \( x^{2} y - x y + 4 x^{2} - 4 x \). Substitute \( y = -4 \) into the expression:\(-4x^2 + 4x + 4x^2 - 4x = 0\). Factoring does not reveal any immediate cancellations. So, we resort to simplifying the expression if possible.
03

Identify Paths for Limits

Analyze the function by substituting \( y = -4 \,(or \, -4 + \varepsilon) \) and \( x = 2 \, (or \, 2 + \delta) \). Substitute into the expression, and simplify it to see if any common terms can lead to zero in both numerator and denominator.
04

Evaluate the Limit Along Different Paths

Calculate the limits along different linear paths. First choose paths like when \( y = -4 \). The expression becomes undefined suggesting looking for a dominant form or factor commonality.
05

Use the Limited Paths for Verification

Attempt different approaches by manipulating the denominator through assumed paths or postulated directions of approach. Confirm through analytic computations, algebra manipulation or l'Hopital's rule; continuously check simplification.

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.

Multivariable Calculus
Multivariable calculus extends the principles of single-variable calculus to functions with multiple variables. This field is essential for understanding how variables interact within a given domain and how changes affect outcomes. In our exercise, we are examining a function of two variables, \(f(x, y)\). Evaluating such functions requires approaching a point from various directions because inputs are no longer limited to a single dimension.Understanding multidimensional spaces is crucial. It ensures that factors like continuity, differentiability, and integrability are accurately applied in multiple dimensions. Additionally, it distinguishes between scalar fields and vector fields, helping solve real-world problems like fluid dynamics, electromagnetism, and graphing surfaces.Exploring these concepts rigorously helps reinforce foundational mathematical skills, paving the way for advanced numerical solution techniques.
Finding Limits
In calculus, finding limits refers to identifying the value that a function approaches as the input approaches some point. For functions of two or more variables, this process can be more challenging than single-variable limits.To find limits in multivariable functions:
  • We approach a given point \((a, b)\) along different paths in the domain.
  • The limit should be the same along all paths for it to exist.
  • An undefined result or discrepancy usually indicates that the limit does not exist.
Understanding how to analyze these paths gives insights into how functions behave at node points. It ensures consistency across solutions, fostering a strong grasp in students of when and how multivariable limits exist.This is key to mastering topics like partial derivatives and double integrals in more advanced courses.
Limit Evaluation Techniques
Several techniques help with evaluating multivariable limits, crucial when direct substitution does not work due to indeterminate forms. Here are some common strategies:1. **Direct Substitution**: When possible, substitute the values directly into the function. If initial substitution leads to a meaningful result without undefined operations, we have found our limit.2. **Simplification or Factorization**: Factor complex numerators and denominators to cancel out common terms. This often simplifies expressions to prevent division by zero or other indeterminate cases.3. **Path Analysis**: Study the limit behavior along different paths. Inconsistent results suggest examining the function more closely for further insights.4. **L'Hopital's Rule**: Apply this for indeterminate forms such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) when applicable.These techniques are foundational tools in assessing the limits of multivariable functions. Mastery involves practicing different scenarios to determine the most efficient technique for specific functions.

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

a. Around the point \((1,0),\) is \(f(x, y)=x^{2}(y+1)\) more sensitive to changes in \(x\) or to changes in \(y\) ? Give reasons for your answer. b. What ratio of \(d x\) to \(d y\) will make \(d f\) equal zero at \((1,0) ?\)

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function \(h=f-\lambda_{1} g_{1}-\lambda_{2} g_{2},\) where \(f\) is the function to optimize subject to the constraints \(g_{1}=0\) and \(g_{2}=0\) . b. Determine all the first partial derivatives of \(h\) , including the partials with respect to \(\lambda_{1}\) and \(\lambda_{2},\) and set them equal to \(0 .\) c. Solve the system of equations found in part (b) for all the unknowns, including \(\lambda_{1}\) and \(\lambda_{2}\) . d. Evaluate \(f\) at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise. Minimize \(f(x, y, z, w)=x^{2}+y^{2}+z^{2}+w^{2}\) subject to the constraints \(\quad 2 x-y+z-w-1=0 \quad\) and \(\quad x+y-z+\) \(w-1=0.\)

Three variables Let \(w=f(x, y, z)\) be a function of three independent variables and write the formal definition of the partial derivative \(\partial f / \partial z\) at \(\left(x_{0}, y_{0}, z_{0}\right) .\) Use this definition to find \(\partial f / \partial z\) at \((1,2,3)\) for \(f(x, y, z)=x^{2} y z^{2}\)

In Exercises \(51-54,\) verify that \(w_{x y}=w_{y x}\) $$w=e^{x}+x \ln y+y \ln x$$

Minimum distance to the origin Find the point closest to the origin on the curve of intersection of the plane \(2 y+4 z=5\) and the cone \(z^{2}=4 x^{2}+4 y^{2}.\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.