Problem 106 - Calculus Volume 3

Question

True or False: If we evaluate \(\lim _{(x, y) \rightarrow(0,0)} f(x)\) along several paths and each time the limit is 1 , we can conclude that \(\lim _{(x, y) \rightarrow(0,0)} f(x)=1\).

Step-by-Step Solution

Answer

False, evaluating limits along several paths does not guarantee a unique limit.

Step 1: Understanding the Problem

We're asked if evaluating the limit \(\lim _{(x, y) \rightarrow(0,0)} f(x)\) along several different paths, with each resulting in 1, is enough to conclude that the limit of \ f(x) \ as \(x, y \rightarrow (0,0)\) is actually 1.

Step 2: Limits along Different Paths

For a function \(f(x, y)\), if the limit changes with different paths, then the overall limit does not exist. Conversely, having the same limit along multiple paths does not guarantee that the overall limit exists.

Step 3: Counterexample Creation

An example can illustrate this: consider \(f(x, y) = \frac{2x^2y}{x^2 + y^2}\). As \(x, y \rightarrow (0,0)\) along the x-axis, y = 0, \(f(x,0) = 0\), and along the y-axis, x = 0, \(f(0,y) = 0\). However, along the path y = x, \(f(x,x) = 1\). The limit changes depending on the path chosen.

Step 4: Conclusion on the Problem

Therefore, it's important to understand that matching results along several paths does not certify the overall limit exists or is that value. We need to ensure the limit is the same regardless of the path, and path-based evaluation does not suffice if those paths aren't exhaustive.

Key Concepts

Limits and Continuity

In multivariable calculus, understanding limits and continuity is essential. When we talk about limits of a function of two variables, we look to see what happens to the function as the point \((x, y)\) approaches a particular point, such as the origin \((0, 0)\). If the function approaches a single value from any direction, we say the limit exists, indicating continuity at that point.
For a function \(f(x, y)\), having this property everywhere means the function is continuous throughout a certain region.
Continuity ensures smooth behavior without sudden jumps or holes anywhere in that region.

• For limits to exist, the approach paths should not affect the result.
• Continuity reinforces the need for consistent outcomes from all directions.

That means no matter how \((x, y)\) reaches \((0, 0)\), the result should consistently be the same.

Path-Dependent Limits

Path-dependent limits are a fascinating aspect of multivariable functions. It is possible for a function to suggest differing limits when approached along different paths. So, what does this mean?

• It implies that simply checking a few paths might be misleading.
• Functions might seem to converge when approached via several paths, but this can be deceptive.

For example, if you evaluate the function \(f(x, y) = \frac{2x^2y}{x^2 + y^2}\) along the x-axis, or the y-axis, you find it provides consistent results, zero in this case. However, if approached along the path \(y = x\), the function yields a limit of 1.
This disparity implies the limit is dependent on the path, thus, the overall limit doesn't exist.
Hence, path-dependent limits are tricky because non-consistent outcomes across various approaches challenge the presence of a single limit.

Counterexample in Calculus

In calculus, counterexamples are incredibly useful tools to show the limitations and boundaries of mathematical rules. They help showcase situations where a seemingly plausible assumption doesn't hold true.
For instance, revisiting our earlier example with \(f(x, y) = \frac{2x^2y}{x^2 + y^2}\), we can clearly see that even though along certain paths the limit seems constant, another path (like \(y = x\)) disputes this constancy by showing a different limit.

• Counterexamples challenge the assumption by offering proof against it.
• They illustrate the necessity for checking broader conditions.

Counterexamples, therefore, emphatically highlight that understanding a function's limit requires thorough examination from all potential directions. Only when every conceivable path to a point shows the same limit can we assert its existence with confidence.