Question

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

Short answer

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

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.

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.

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. \(y=x^2\)

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.

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 \(\frac{1}{2}\).

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.

Key concepts

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’s 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.