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Chapter 13: Problem 76

Show that $$\lim _{(x, y) \rightarrow(0,0)} \frac{a x^{2(p-n)} y^{n}}{b x^{2 p}+c y^{p}} \text { does }$$ not exist when \(a, b,\) and \(c\) are nonzero real numbers and \(n\) and \(p\) are positive integers with \(p \geq n\)

Short Answer

Expert verified
Question: Prove that the limit does not exist for the given function: $$ \lim_{(x,y) \to (0,0)} \frac{a x^{2(p-n)} y^{n}}{b x^{2p}+c y^{p}} $$ where \(a, b, c\) are nonzero real numbers, \(n\) and \(p\) are positive integers, and \(p \geq n\). Answer: The limit does not exist because we found two different paths, with different values of \(k\), that give different limit values. For \(k=1\), the limit is \(\frac{a}{b}\), and for \(k=2\), the limit is \(\frac{a}{b+c}\). Since these limits are not equal, the limit does not exist when approaching \((0,0)\).

Step by step solution

01

Rewrite the given expression in a more convenient form

Let's rewrite the given expression to easily perform further operations: $$ L = \frac{a x^{2(p-n)} y^{n}}{b x^{2p}+c y^{p}} $$
02

Analyze the given constraints

We are given that \(a\), \(b\), and \(c\) are nonzero real numbers, \(n\) and \(p\) are positive integers, and \(p \geq n\).
03

Find a suitable path and limits along the path

Let's consider the path \(y = x^k\), where \(k\) is a positive integer. We can later try different values of \(k\) to find two paths that yield different limit values: $$ L = \frac{a x^{2(p-n)} (x^k)^{n}}{b x^{2p}+c (x^k)^{p}} $$ Now, we'll simplify the expression: $$ L = \frac{a x^{2p}}{b x^{2p}+c x^{kp}} $$ As \((x, y) \rightarrow (0,0)\), \(x \rightarrow 0\). Hence, by applying the limit, we get: $$ \lim_{x \rightarrow 0} L = \lim_{x \rightarrow 0} \frac{a x^{2p}}{b x^{2p}+c x^{kp}} = \lim_{x \rightarrow 0} \frac{ax^{2p}}{bx^{2p}+\frac{cx^{kp}}{ax^{2p}}} \cdot \frac{1}{x^{2p}} $$ Simplifying further, we get: $$ \lim_{x \rightarrow 0} L = \frac{a}{b+\frac{c}{x^{(k-2)p}}} $$
04

Find two different limits using different values of \(k\)

Let's evaluate the limit using different values of \(k\): 1. When \(k=1\), the limit becomes: $$ \lim_{x \rightarrow 0} L = \frac{a}{b+\frac{c}{x^{(1-2)p}}} = \frac{a}{b} \quad(\text{since } p \geq n \Rightarrow (1-2)p \leq 0) $$ 2. When \(k=2\), the limit becomes: $$ \lim_{x \rightarrow 0} L = \frac{a}{b+\frac{c}{x^{(2-2)p}}} = \frac{a}{b+c} \quad(\text{since } p \geq n \Rightarrow (2-2)p=0) $$ Since we have found two different paths (values of \(k\)) that give different limit values (which are not equal), the limit does not exist.

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Key Concepts

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

Limits in Multiple Variables
When dealing with limits in multiple variables, we are often interested in understanding how a function behaves as it approaches a particular point in a two-dimensional or higher-dimensional space. For instance, in our exercise, we approach the origin (0, 0) in a two-dimensional plane.
In single variable calculus, you can only approach a point in two directions: from the left and from the right. However, in multivariable calculus, there are infinitely many directions you can approach a point. This makes analyzing limits in multiple variables more complex.

To study these limits, it's crucial to substitute different paths and observe if the function's limit remains constant as you approach the target point. If the limit is the same regardless of the path taken, then the limit at that point exists. If not, it does not exist.
Path Dependence in Limits
Path dependence in limits is a concept that arises specifically in the context of multivariable limits. It indicates that the limit of a function can vary depending on the path taken to approach a particular point.

For example, in our exercise, we used the paths where \(y = x^k\), allowing us to substitute different values for \(k\). We observed different limit values along these paths.
  • When \(k=1\), the limit simplifies to \(\frac{a}{b}\).
  • When \(k=2\), the limit changes to \(\frac{a}{b+c}\).
Since these two paths yield different limit values, the limit does not exist as a unique value.
This showcases the importance of checking multiple paths to determine whether multivariable limits exist or differ based on the path taken. By revealing different values, path dependence proves that a function鈥檚 behavior is not uniform across the approaching paths.
Nonexistence of Limits in Calculus
In calculus, demonstrating the nonexistence of a limit usually involves showing that the function does not approach a single value as it nears a certain point.

For multivariable functions, one common method is to find two or more paths that result in distinct limits, as was done in our given problem.
This evidence of path-dependent limits proves that the limit cannot settle onto a single value.

Nonexistence is critical to recognize because it indicates that the behavior of the function near a point is unpredictable or discontinuous.
  • The different values along selected paths signify the absence of a unique limit.
  • It helps in understanding and categorizing functions based on continuity and differentiability.
The concept of nonexistence plays a significant role in higher-level calculus, as it guides us in understanding the intricacies and limitations of behavior within the multiple dimensions of functions.

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Most popular questions from this chapter

Find the point on the surface curve \(y=x^{2}\) nearest the line \(y=x-1 .\) Identify the point on the line.

Let \(w=f(x, y, z)=2 x+3 y+4 z\), which is defined for all \((x, y, z)\) in \(\mathbb{R}^{3}\). Suppose that we are interested in the partial derivative \(w_{x}\) on a subset of \(\mathbb{R}^{3}\), such as the plane \(P\) given by \(z=4 x-2 y .\) The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane \(P\), consider \(x\) and \(y\) as the independent variables, which means \(z\) depends on \(x\) and \(y,\) so we write \(w=f(x, y, z(x, y)) .\) Differentiate with respect to \(x\) holding \(y\) fixed to show that \(\left(\frac{\partial w}{\partial x}\right)_{y}=18,\) where the subscript \(y\) indicates that \(y\) is held fixed. b. Alternatively, on the plane \(P,\) we could consider \(x\) and \(z\) as the independent variables, which means \(y\) depends on \(x\) and \(z,\) so we write \(w=f(x, y(x, z), z)\) and differentiate with respect to \(x\) holding \(z\) fixed. Show that \(\left(\frac{\partial w}{\partial x}\right)_{z}=8,\) where the subscript \(z\) indicates that \(z\) is held fixed. c. Make a sketch of the plane \(z=4 x-2 y\) and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x}\), \(\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\).

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$

Use the definition of the gradient (in two or three dimensions), assume that \(f\) and \(g\) are differentiable functions on \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3},\) and let \(c\) be a constant. Prove the following gradient rules. a. Constants Rule: \(\nabla(c f)=c \nabla f\) b. Sum Rule: \(\nabla(f+g)=\nabla f+\nabla g\) c. Product Rule: \(\nabla(f g)=(\nabla f) g+f \nabla g\) d. Quotient Rule: \(\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}\) e. Chain Rule: \(\nabla(f \circ g)=f^{\prime}(g) \nabla g,\) where \(f\) is a function of one variable
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