Problem 86 - 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. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}\) a. Along the \(x\) -axis \((y=0)\) b. Along the \(y\) -axis \((x=0)\) c. Along the path \(y=2 x\)

Step-by-Step Solution

Answer

The limit does not exist because it depends on the path taken.

Step 1: Evaluate Along the x-axis (y=0)

Substitute \( y = 0 \) into the function. The expression becomes:\[\lim_{(x, y) \rightarrow (0, 0)} \frac{x \cdot 0 + 0^3}{x^2 + 0^2} = \lim_{x \rightarrow 0} \frac{0}{x^2}\]The numerator is zero, leading to the limit:\[0\].

Step 2: Evaluate Along the y-axis (x=0)

Substitute \( x = 0 \) into the function. The expression becomes:\[\lim_{(x, y) \rightarrow (0, 0)} \frac{0 \cdot y + y^3}{0^2 + y^2} = \lim_{y \rightarrow 0} \frac{y^3}{y^2} = \lim_{y \rightarrow 0} y \]Which evaluates to:\[0\].

Step 3: Evaluate Along the Path y=2x

Substitute \( y = 2x \) into the function. The expression changes to:\[\lim_{(x, y) \rightarrow (0, 0)} \frac{x(2x) + (2x)^3}{x^2 + (2x)^2} = \lim_{x \rightarrow 0} \frac{2x^2 + 8x^3}{x^2 + 4x^2}\]Which simplifies to:\[\lim_{x \rightarrow 0} \frac{2x^2 + 8x^3}{5x^2} = \lim_{x \rightarrow 0} \frac{2 + 8x}{5}\]Since \( x \rightarrow 0 \), the limit evaluates to \( \frac{2}{5} \).

Step 4: Determine if the Limit Exists

Since the limits differ for different paths (0 for the x-axis and y-axis, \( \frac{2}{5} \) for \( y = 2x \)), we conclude the limit does not exist as it approaches different values depending on the path.

Key Concepts

Limits in Multiple Variables

In multivariable calculus, limits play a crucial role in understanding the behavior of functions as they approach particular points in the domain. This is very similar to the single-variable case but with added complexity. When dealing with functions of two or more variables, the limit must be considered from every possible direction or path leading to the point of interest. For example, consider the function given in the exercise, \[\lim _{(x, y) \rightarrow(0,0)} \frac{x y+y^{3}}{x^{2}+y^{2}}.\]In this case, the limit is evaluated at the origin \((0,0)\).

• The critical point here is to understand that if a limit is to exist for a multivariable function, the function must approach the same value irrespective of the direction from which the point is approached.
• If the evaluated limit changes when approached along different paths, then the limit is said to be nonexistent.

Understanding the concept of limits in multiple variables is foundational for more advanced topics such as continuity and differentiability in multivariable calculus.

Evaluating Limits Along Paths

To determine if a limit exists for functions of several variables, one commonly used method is path evaluation. It involves checking the limit of the function along various specific paths leading to the target point. This technique provides insight into how the function behaves from different approaches. In the provided exercise:

• First, the limit was evaluated along the x-axis by setting \(y = 0\), which yielded a limit of \(0\).
• Next, along the y-axis, with \(x = 0\), the limit also resulted in \(0\).
• However, when evaluated along the path \(y = 2x\), the value was different, specifically \(\frac{2}{5}\).

Evaluating along several paths might give the same limit value, indicating the potential existence of a limit. However, differing limit values, such as in this problem, indicate the nonexistence of the limit at that point.

Nonexistent Limits

In mathematics, a nonexistent limit occurs when a function approaches different values along various paths leading to a single point. This scenario is particularly common in functions of multiple variables where the limit's behavior can vary vastly depending on the approach direction. For the example function from the exercise, the approach paths along the x-axis and y-axis both yielded a limit of \(0\). In contrast, the path \(y = 2x\) led to a limit of \(\frac{2}{5}\). These discrepancies highlight the concept of nonexistent limits in multivariable calculus, which often leads to fascinating mathematical behavior.

• The key takeaway is that for a limit to be existent, it must approach the same value from every possible path to the point.
• If different paths yield different limit values, the overall limit at that point is said to not exist.

Understanding this concept is essential because it underlines the importance of directionality in multivariable calculus, impacting subsequent mathematical analysis and applications.