Problem 88 - Calculus Volume 3

Question

For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y}{x^{4}+y^{2}}\) a. Along the \(x\) -axis \((y=0)\) b. Along the \(y\) -axis \((x=0)\) c. Along the path \(y=x^{2}\)

Step-by-Step Solution

Answer

The limit does not exist because different paths yield different limit values.

Step 1: Evaluate Limit Along the x-axis

Substitute \(y=0\) into the function to evaluate the limit along the x-axis: \[\lim_{(x,0) \to (0,0)} \frac{x^{2} \cdot 0}{x^{4}+0^2} = \lim_{x \to 0} \frac{0}{x^{4}} = 0.\]Hence, the limit along the x-axis is 0.

Step 2: Evaluate Limit Along the y-axis

Substitute \(x=0\) into the function to find the limit along the y-axis: \[\lim_{(0,y) \to (0,0)} \frac{0^2 \cdot y}{0^4+y^{2}} = \lim_{y \to 0} \frac{0}{y^{2}} = 0.\]Thus, the limit along the y-axis is 0.

Step 3: Evaluate Limit Along the Path \(y=x^2\)

Substitute \(y = x^2\) into the function: \[\lim_{(x,x^2) \to (0,0)} \frac{x^{2} \cdot x^2}{x^{4}+(x^2)^{2}} = \lim_{x \to 0} \frac{x^{4}}{x^{4}+x^{4}} = \lim_{x \to 0} \frac{x^{4}}{2x^{4}} = \lim_{x \to 0} \frac{1}{2} = \frac{1}{2}.\]Therefore, the limit along the path \(y=x^2\) is \(\frac{1}{2}\).

Step 4: Conclusion

Since the limit values differ along different paths (0 along the x-axis and y-axis, and \(\frac{1}{2}\) along the path \(y=x^2\)), the overall limit does not exist. For a limit to exist, the value must be the same along all paths approaching the point.

Key Concepts

Limit along a path

When analyzing limits in multivariable calculus, examining a limit along various paths is crucial. This approach helps understand how the function behaves as it approaches a specific point from different directions. In the exercise, we looked at the function \(\frac{x^2 y}{x^4 + y^2}\) and evaluated its limit as \((x, y)\) approaches \((0,0)\) along different paths: the x-axis, y-axis, and the path \(y = x^2\).

By substituting specific paths into the function, we can see how the limit behaves:

• Along the x-axis, set \(y = 0\), and the limit becomes 0.
• Along the y-axis, set \(x = 0\), and the limit is also 0.
• However, along the path \(y = x^2\), the limit turns out to be \(\frac{1}{2}\).

This difference in results along distinct paths is a strong indicator to check whether a multivariable limit truly exists.

Limit does not exist

In multivariable calculus, a limit does not exist if a function approaches different values along different paths towards the same point. Using the provided example, let's see why the overall limit does not exist for the function \(\frac{x^2 y}{x^4 + y^2}\) as \((x, y)\) approaches \((0,0)\).

The calculations along the x-axis, y-axis, and path \(y = x^2\) resulted in:

• 0 along the x-axis
• 0 along the y-axis
• \(\frac{1}{2}\) along the path \(y = x^2\)

Since the results are not the same, we conclude that the limit does not exist. For the limit to be defined, it must yield the same value through all possible paths or directions approaching the point in question.

Evaluating limits

Evaluating multivariable limits involves checking the behavior of functions as they approach certain points from different paths in a multi-dimensional space. The primary strategy is to substitute the path equations into the function and simplify to find the resulting limit as the variable approaches the point of interest.
Here’s how we typically evaluate limits:

• Specify the paths, such as the x-axis, y-axis, or more complex paths like \(y = x^2\).
• Substitute these paths into the given function.
• Simplify the expression and calculate the limit.

This systematic approach helps determine whether the function has a consistent limit regardless of the path taken toward the point. In cases where the limits differ, it indicates the function's behavior is inconsistent and thus the limit does not exist.

Approaching a point

The concept of "approaching a point" is fundamental in understanding limits in calculus, particularly in the multivariable context. This involves considering how a variable set, in this case, \((x, y)\), gets infinitely close to a specific point, such as \((0, 0)\).
In multivariable limits, unlike single-variable limits, there are infinitely many paths through which a point can be approached:

• The simplest paths are usually the coordinate axes (e.g., x-axis and y-axis).
• Various curves or lines, such as \(y = x\) or \(y = x^2\).

This method highlights whether a consistent value is being approached by the function regardless of the path. If different paths yield different limit values, it implies divergent behaviors of the function at that point, meaning the limit does not exist. Therefore, understanding how a function's values change as it approaches a point from various directions allows us to determine the existence of a limit.