/*! 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 21 Determine whether the limit exis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 21

Determine whether the limit exists. If so, find its value. $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{e^{\sqrt{x^{2}+y^{2}+z^{2}}}}{\sqrt{x^{2}+y^{2}+z^{2}}} $$

Short Answer

Expert verified
The limit exists and is equal to 1.

Step by step solution

01

Understand the Function and Limit Point

We are asked to find \( \lim_{(x, y, z) \rightarrow (0,0,0)} \frac{e^{\sqrt{x^{2}+y^{2}+z^{2}}}}{\sqrt{x^{2}+y^{2}+z^{2}}} \). The limit point is the origin (0,0,0) in three-dimensional space, and the function involves an exponential term divided by a radical expression. We need to determine whether this limit exists as \((x, y, z)\) approaches zero.
02

Convert to Spherical Coordinates

Since the limit involves \(x^2 + y^2 + z^2\), it's beneficial to convert to spherical coordinates where \(\rho = \sqrt{x^2 + y^2 + z^2}\), which simplifies the expression. This transforms the limit to \(\lim_{\rho \rightarrow 0} \frac{e^{\rho}}{\rho}\).
03

Analyze the Limit Expression

Look at the expression \( \lim_{\rho \rightarrow 0} \frac{e^{\rho}}{\rho} \). Direct substitution is not possible because it leads to the indeterminate form \(\frac{1}{0}\), which suggests further analysis is needed or the use of L'H么pital's Rule if applicable.
04

Apply L'H么pital's Rule

L'H么pital's Rule states that for limits of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), we can take the derivative of the numerator and the denominator. In this case, we take the derivative of \(e^\rho\) and \(\rho\), which gives us: \[\lim_{\rho \rightarrow 0} \frac{\frac{d}{d\rho}(e^{\rho})}{\frac{d}{d\rho}(\rho)} = \lim_{\rho \rightarrow 0} \frac{e^{\rho}}{1} = \lim_{\rho \rightarrow 0} e^{\rho}.\]
05

Evaluate the Simplified Limit

The simplified limit expression \(\lim_{\rho \rightarrow 0} e^{\rho}\) is direct, and since \(e^{\rho}\) approaches \(e^0 = 1\) as \(\rho \rightarrow 0\), we find that the limit is equal to 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.

Limit of a function
The concept of the limit of a function, especially in multivariable calculus, is essential to understanding how functions behave as their inputs approach certain points. In our exercise, we are interested in the behavior of a function as its variables
  • Approach a specific point, in this case, the origin (0, 0, 0).
  • The key question is whether the function approaches a particular value when getting infinitely close to that point.
For the given function, the challenge is to ascertain the limit as
  • \((x, y, z) \to (0, 0, 0)\).
By understanding limits, you determine the point's value that the function is heading towards. However, in multivariable scenarios, approach paths matter. Approaching from different 3D directions could result in different limits. Thus, it's crucial to consider all paths. When the limit is the same despite the path, it implies the existence of the limit.
Spherical coordinates
Spherical coordinates can greatly simplify problems in three-dimensional space involving radial symmetry, like in our function where you see
  • The expression \(x^2 + y^2 + z^2\).
In spherical coordinates:
  • \(\rho\) represents the radial distance from the origin (similar to radius in polar coordinates),
  • \(\theta\) is the azimuthal angle in the xy-plane from the x-axis, and
  • \(\phi\) denotes the angle from the z-axis.
By converting to spherical coordinates where \(\rho = \sqrt{x^2 + y^2 + z^2}\), our original problem reduces to focusing solely on \(\rho\), simplifying the limit analysis. This makes it easier to visualize and mathematically handle the path to the origin, focusing on how the function behaves as the distance to the origin, \(\rho\), shrinks.
L'H么pital's Rule
L'H么pital's Rule is a powerful tool in calculus for finding limits of indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In our problem, the transformed limit expression using spherical coordinates, \(\lim_{\rho \to 0} \frac{e^{\rho}}{\rho}\), initially presents an indeterminate form. Here, L'H么pital's Rule comes into play:
  • By differentiating both the numerator \(e^\rho\) and the denominator \(\rho\).
  • We turn the expression into \(\lim_{\rho \to 0} \frac{e^{\rho}}{1}\).
This transformation makes the limit straightforward, giving \(\lim_{\rho \to 0} e^\rho = e^0 = 1\). L'H么pital's Rule enables us to evaluate limits that would otherwise be overly complex or undefined at first glance while confirming the validity of our mathematical intuition by simplifying challenging expressions.

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 for the tangent plane and parametric equations for the normal line to the surface at the point \(P\). $$ z=\ln \sqrt{x^{2}+y^{2}} ; P(-1,0,0) $$

The temperature (in degrees Celsius) at a point \((x, y, z)\) in a metal solid is $$ T(x, y, z)=\frac{x y z}{1+x^{2}+y^{2}+z^{2}} $$ (a) Find the rate of change of temperature with respect to distance at \((1,1,1)\) in the direction of the origin. (b) Find the direction in which the temperature rises most rapidly at the point \((1,1,1)\). (Express your answer as a unit vector.) (c) Find the rate at which the temperature rises moving from \((1,1,1)\) in the direction obtained in part (b).

Given that the directional derivative of \(f(x, y, z)\) at the point \((3,-2,1)\) in the direction of \(\mathbf{a}=2 \mathbf{i}-\mathbf{j}-2 \mathbf{k}\) is \(-5\) and that \(\|\nabla f(3,-2,1)\|=5\), find \(\nabla f(3,-2,1)\)

Find the absolute extrema of the given function on the indicated closed and bounded set \(R .\) \(f(x, y)=x y-x-3 y ; R\) is the triangular region with vertices \((0,0),(0,4)\), and \((5,0)\).

Find an equation for the tangent plane and parametric equations for the normal line to the surface at the point \(P\). $$ z=x e^{-y} ; P(1,0,1) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

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.