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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).

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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.

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Most popular questions from this chapter

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

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Are the following statements true or false? (a) The gradient vector \(\nabla f(a, b)\) is tangent to the contour of \(f\) at \((a, b)\). (b) \(f_{\vec{u}}(a, b)=\|\nabla f(a, b)\| .\) (c) \(f_{\vec{u}}(a, b)\) is parallel to \(\vec{u}\). (d) If \(\vec{u}\) is perpendicular to \(\nabla f(a, b),\) then \(f_{\vec{u}}(a, b)=\langle 0,0\rangle\). (e) If \(\vec{u}\) is a unit vector, then \(f_{\vec{u}}(a, b)\) is a vector. (f) Suppose \(f_{x}(a, b)\) and \(f_{y}(a, b)\) both exist. Then there is always a direction in which the rate of change of \(f\) at \((a, b)\) is zero. (g) If \(f(x, y)\) has \(f_{x}(a, b)=0\) and \(f_{y}(a, b)=0\) at the point \((a, b)\), then \(f\) is constant everywhere. (h) \(\nabla f(a, b)\) is a vector in 3 -dimensional space.

In the following questions, we determine and apply the linearization for several different functions. a. Find the linearization \(L(x, y)\) for the function \(f\) defined by \(f(x, y)=\) \(\cos (x)\left(2 e^{2 y}+e^{-2 y}\right)\) at the point \(\left(x_{0}, y_{0}\right)=(0,0) .\) Use the linearization to estimate the value of \(f(0.1,0.2)\). Compare your estimate to the actual value of \(f(0.1,0.2)\) b. The Heat Index, \(I,\) (measured in apparent degrees \(\mathrm{F}\) ) 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, \(I=I(T, H),\) is provided in Table 10.4 .13 $$\begin{array}{ccccc}\hline T \downarrow \backslash H \rightarrow & 70 & 75 & 80 & 85 \\ \hline 90 & 106 & 109 & 112 & 115 \\\\\hline 92 & 112 & 115 & 119 & 123 \\ \hline 94 & 118 & 122 & 127 & 132 \\\\\hline 96 & 125 & 130 & 135 & 141 \\\\\hline\end{array}$$ Suppose you are given that \(I_{T}(94,75)=3.75\) and \(I_{H}(94,75)=0.9\). Use this given information and one other value from the table to estimate the value of \(I(93.1,77)\) using the linearization at (94,75) . Using proper terminology and notation, explain your work and thinking. c. Just as we can find a local linearization for a differentiable function of two variables, we can do so for functions of three or more variables. By extending the concept of the local linearization from two to three variables, find the linearization of the function \(h(x, y, z)=e^{2 x}(y+\) \(z^{2}\) ) at the point \(\left(x_{0}, y_{0}, z_{0}\right)=(0,1,-2) .\) Then, use the linearization to estimate the value of \(h(-0.1,0.9,-1.8)\).

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.
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