/*! 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 2 In Problems 1-14, use the proper... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 2

In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(-1,1)}\left(2 x y+3 x^{2}\right) $$

Short Answer

Expert verified
The limit is 1.

Step by step solution

01

Identify the Limit Expression

We need to calculate the limit of the expression \( 2xy + 3x^2 \) as \((x, y)\) approaches \((-1, 1)\). First, recognize that this is a polynomial expression, involving \(x\) and \(y\).
02

Apply the Direct Substitution Property

Since the expression \(2xy + 3x^2\) is a polynomial, limits can be directly substituted according to the limit properties for polynomials. Substitute \(x = -1\) and \(y = 1\) into the expression.
03

Evaluate the Substituted Expression

Calculate the result of substituting the given coordinates into the polynomial. Substitute and compute: \(2(-1)(1) + 3(-1)^2\).
04

Simplify the Expression

Now, simplify the expression from Step 3. \(2(-1)(1) = -2\) and \(3(-1)^2 = 3\). Add these results: \(-2 + 3 = 1\).
05

State the Result of the Limit

The calculation results in \(1\). Therefore, the limit of the expression as \((x, y)\) approaches \((-1, 1)\) is 1.

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.

Understanding Limits
Limits in calculus help us understand how a function behaves as its inputs approach a specific value. Imagine trying to find the value a function is "heading towards" without necessarily reaching it.
In this exercise, we aim to determine what happens to the function \(2xy + 3x^2\) as \((x, y)\) gets closer and closer to \((-1, 1)\).
  • A limit captures how output values of a function approach a single number as input variables get arbitrarily close to a specific point.
  • This concept is crucial for calculus, particularly in the context of continuity and derivatives.
  • Understanding and applying limits, like in our exercise, becomes easier when you know whether direct substitution can be applied or further manipulation is needed.
This prepares us for solving the limit using its properties in polynomial functions.
Polynomial Functions
Polynomial functions are a specific type of mathematical expression consisting of variables and coefficients. These expressions contain terms in the form of constants, powers, and their products. For example, \(2xy + 3x^2\) is a polynomial in terms of \(x\) and \(y\).
Key characteristics of polynomial functions include:
  • They are composed of smooth, curved lines and can be of degree zero (constant) or higher for one or more variables.
  • Polynomials are straightforward to work with because they are continuous everywhere and differentiable.
  • Limits of polynomial functions, such as the one explored in our exercise, can typically be solved by substituting directly.
Understanding these properties allows us to employ direct substitution effectively when appropriate.
Direct Substitution Method
The direct substitution method is a straightforward approach recognized by its simplicity in solving limits when dealing with polynomial functions like the one in our example.
Here's how it works:
  • If the function is polynomial and the variables approach a finite point, you can often substitute the values directly into the function.
  • This approach relies on the property that polynomials are continuous, meaning their limits at any point correspond to their actual values at that point.
  • In our specific exercise, substituting \(x = -1\) and \(y = 1\) directly into \(2xy + 3x^2\) produces a straightforward computation which can be simplified to find the limit.
By recognizing this ease, we simplify our workflow in calculus, enabling us to solve problems with confidence and efficiency.

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 \(\frac{d y}{d x}\) if \(y=\arctan x\)Find \(\frac{d y}{d x}\) if \(y=\arctan x\).

In what direction does \(f(x, y)=e^{x} \cos y\) increase most rapidly at \((0, \pi / 2) ?\)

Find a unit vector that is normal to the level curve of the function $$f(x, y)=x^{2}-y^{3}$$ at the point \((1,3)\).

Show that the equilibrium \(\left[\begin{array}{l}0 \\ 0\end{array}\right]\) of $$ \left[\begin{array}{c} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} 2 & -4 \\ 5 & -6 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ is stable.

Find the gradient of each function. $$ f(x, y)=x\left(x^{2}-y^{2}\right)^{2 / 3} $$
See all solutions

Recommended explanations on Biology Textbooks

Ecological Levels

Read Explanation

Astrobiological Science

Read Explanation

Biological Structures

Read Explanation

Cell Cycle

Read Explanation

Control of Gene Expression

Read Explanation

Responding to Change

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.