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Chapter 15: Problem 5

Explain how examining limits along multiple paths may prove the nonexistence of a limit.

Short Answer

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Question: Explain the concept of examining limits along multiple paths and how it may prove the nonexistence of a limit. Answer: Examining limits along multiple paths involves calculating the limit of a multivariable function as the input approaches a certain point from different directions. If the limit exists, it should have the same value regardless of the path chosen. However, if we find at least two different paths that provide different limits, then it proves the nonexistence of a limit. This method helps us to confirm whether a limit exists or not for a given function at a specific point.

Step by step solution

01

Understand the Concept of Limits

The limit of a multivariable function at a certain point refers to the value that the function approaches as the input approaches that point. If the limit exists, it should have the same value regardless of the path taken by the inputs.
02

Explain the Basic Strategy for Examining Limits Along Multiple Paths

We can examine limits along multiple paths by calculating the limit of a function as the input approaches a point from different directions. If all these limits are the same, it suggests that the limit exists. However, if we can find two paths that provide different limits, then it proves the nonexistence of a limit.
03

Find an Example to Show the Nonexistence of a Limit

Consider the following function: \[ f(x,y)=\frac{x^2y}{x^4+y^2} \] We will find the limit of the function as \((x,y)\) approaches \((0,0)\) along different paths. Choosing these two paths: 1. \(y=0\) 2. \(y=x^2\)
04

Calculate the Limit Along the First Path

If we take the first path and set \(y=0\), our function becomes: \[ f(x,0)=\frac{x^2(0)}{x^4+0} = 0 \] Thus, along the path \(y=0\), the limit of the function as \((x,y)\) approaches \((0,0)\) is 0.
05

Calculate the Limit Along the Second Path

For the second path, we set \(y=x^2\), the function becomes: \[ f(x,x^2)=\frac{x^2(x^2)}{x^4+x^4} = \frac{x^4}{2x^4} = \frac{1}{2} \] Therefore, the limit of the function as \((x,y)\) approaches \((0,0)\) along the path \(y=x^2\) is \(\frac{1}{2}\).
06

Conclude the Nonexistence of a Limit

Since we found two different limits along two different paths (i.e., 0 and \(\frac{1}{2}\)), this proves that the limit of the function as \((x,y)\) approaches \((0,0)\) does not exist. By examining limits along multiple paths, we can confirm the nonexistence of a limit.

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Key Concepts

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

limit along paths
When dealing with multivariable functions, one key strategy to determine the existence of a limit is examining how limits behave along different paths towards a particular point.
This method is particularly useful because the behavior of the function may vary depending on the direction from which you approach.
To find these limits, we can select specific paths, such as straight lines or curves, and evaluate if the limit remains consistent.
  • Consider using simple paths first, such as horizontal, vertical, or diagonal lines. These are usually straightforward to calculate and provide clear insights.
  • If the limits along all paths are the same, it strongly suggests that a limit exists at that point.
  • Different limits indicate a discrepancy, suggesting further investigation into the function鈥檚 behavior is necessary.
In our exercise, by taking the paths where either one coordinate equals zero or a power of the other, you can unmask differences in the limit values.
nonexistence of a limit
The nonexistence of a limit arises when a multivariable function does not approach a single value as the inputs approach a particular point.
For the limit to exist at this point, the function must converge to the same value along every possible path.
  • When differing limits are found, like in the classic scenario we've addressed, this immediately flags the nonexistence of a consistent limit at that chosen point.
  • Using different paths like vertical and parabolic, as done in the original exercise, is a practical approach to reveal such inconsistencies.
In the example given, we derived two distinct limits (0 and \(\frac{1}{2}\)) from different paths. This conflict proves that the overall limit at \(0,0\) does not exist.
Thus, confirming that examining multiple paths is a vital method in verifying the nonexistence of limits in multivariable contexts.
multivariable functions
Multivariable functions are mathematical functions with more than one input variable, commonly seen as \(f(x,y)\) or \(f(x,y,z)\).
These functions can represent surfaces or more complex geometrical shapes in multi-dimensional spaces.
  • Understanding these functions involves visualizing them as surfaces or curves in 3D space, which can be complex due to the additional dimensions.
  • Analyzing limits in this context often requires examining various pathways to a point because the behavior of these functions can differ vastly depending on the direction of approach.
In the exercise, the function \(f(x,y)=\frac{x^2y}{x^4+y^2}\) is explored. The limit's nonexistence at \(0,0\) highlights how multivariable functions can behave unexpectedly due to their complexity.
This analysis shows the necessity of a comprehensive approach to understanding their behavior thoroughly, especially when confirming characteristics such as continuity or limits.

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Most popular questions from this chapter

Use Lagrange multipliers in the following problems. When the constraint curve is unbounded, explain why you have found an absolute maximum or minimum value. Maximum volume cylinder in a sphere Find the dimensions of the right circular cylinder of maximum volume that can be inscribed in a sphere of radius 16

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, a local minimum, or a saddle point. If the Second Derivative Test is inconclusive, determine the behavior of the function at the critical points. $$f(x, y)=\sin (2 \pi x) \cos (\pi y), \text { for }|x| \leq \frac{1}{2} \text { and }|y| \leq \frac{1}{2}$$

The level curves of the surface \(z=x^{2}+y^{2}\) are circles in the \(x y\) -plane centered at the origin. Without computing the gradient, what is the direction of the gradient at (1,1) and (-1,-1) (determined up to a scalar multiple)?

Consider the following functions \(f,\) points \(P,\) and unit vectors \(\mathbf{u}\). a. Compute the gradient of \(f\) and evaluate it at \(P\). b. Find the unit vector in the direction of maximum increase of \(f\) at \(P\). c. Find the rate of change of the function in the direction of maximum increase at \(P\) d. Find the directional derivative at \(P\) in the direction of the given vector. $$f(x, y, z)=x^{2}+2 y^{2}+4 z^{2}+10 ; P(1,0,4) ;\left\langle\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right\rangle$$

Suppose \(\mathbf{n}\) is a vector normal to the tangent plane of the surface \(F(x, y, z)=0\) at a point. How is \(\mathbf{n}\) related to the gradient of \(F\) at that point?
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