Problem 17 - APEX Calculus

Question

Evaluate the limit along the paths given, then state why these results show the given limit does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y-y^{2}}{y^{2}+x}\) (a) Along the path \(y=m x\). (b) Along the path \(x=0\).

Step-by-Step Solution

Answer

The limit does not exist since different paths yield different results.

Step 1: Evaluate the Limit Along the Path y=mx

Substitute \( y = mx \) into the function: \[\frac{x(mx) - (mx)^2}{(mx)^2 + x} = \frac{mx^2 - m^2x^2}{m^2x^2 + x} \]This simplifies to: \[\frac{mx^2(1-m)}{m^2x^2 + x} \]Divide numerator and denominator by \(x\): \[\frac{mx(1-m)}{m^2x + 1} \]Now, take the limit as \(x\) approaches 0: \[\lim_{x \to 0} \frac{mx(1-m)}{m^2x + 1} = 0\]Thus, the limit along the path \( y = mx \) is 0.

Step 2: Evaluate the Limit Along the Path x=0

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

Step 3: Compare the Different Limits

From Step 1, the limit along \( y = mx \) is 0, while the limit found in Step 2 along \( x = 0 \) is \(-1\). Since different paths result in different limits, the overall limit does not exist at \((0,0)\).

Key Concepts

Limit Paths

In multivariable calculus, when evaluating limits of functions with two or more variables, we often explore different limit paths. This means examining how a function behaves as it approaches a particular point along various lines or curves. This approach helps us understand if a limit exists by checking if the function yields the same result regardless of the path taken.

Consider the function in our problem, \( \lim_{(x, y) \rightarrow(0,0)} \frac{x y-y^{2}}{y^{2}+x} \). To find this limit, we select specific paths. Each path is a trajectory or line along which \( (x, y) \) converges to the point (0,0).

• Linear Paths: Like \( y = mx \), where \( m \) is the slope of the line. This transforms our function into another form that depends on a single variable to simplify evaluation.
• Vertical or Horizontal Paths: Such as \( x = 0 \) or \( y = 0 \), where the function simplifies considerably.

By evaluating these paths, we conclude the limit along each route. These different calculations are crucial to identify whether a consistent limit exists as we approach the target point from any direction.

Limit Evaluation

Limit evaluation involves calculating the limit of a function as it approaches a specific point along a chosen path. Evaluating limits in multivariable calculus can be more complex than in single-variable cases because the function's behavior can differ across various paths.

In our example, the limit is evaluated by substituting specific paths into the function:

• Substituting \( y = mx \): This led to the limit of 0. By replacing \( y \) with \( mx \), we simplified the function and calculated the limit as we let \( x \to 0 \). The process showed that the function equalized to 0 along this path.
• Substituting \( x = 0 \): This led to the limit of -1. By setting \( x \) to 0, the function simplified to \( \frac{-y^2}{y^2} \), which consistently yielded -1 as \( y \to 0 \).

The evaluations highlight how different substitution methods along distinct paths provide the necessary information to understand a function's behavior at a particular point. Understanding these calculations allows one to identify if the limits converge to the same value for all paths.

Nonexistent Limits

Nonexistent limits occur when a function approaches different values along different paths as it converges towards a particular point. In multivariable functions, this concept underscores the complexity of limit evaluations compared to single-variable functions.

If a function's limit exists, the result should be the same no matter which path we choose. However, our example function \( \lim_{(x, y) \rightarrow(0,0)} \frac{x y-y^{2}}{y^{2}+x} \) demonstrated different results:

• The limit along the path \( y = mx \) was 0.
• The limit along the path \( x = 0 \) was -1.

Since these path results weren't identical, it indicates that the limit at \( (0,0) \) does not exist. This diversity in outcomes across various paths is a clear marker of a nonexistent limit, emphasizing that the function does not have a single, consistent value at the point in question.

Recognizing such discrepancies is key in multivariable calculus as it informs us that the behavior of a function can be unpredictable and complex as it approaches certain points. Understanding nonexistent limits is crucial for grasping more advanced concepts in the mathematical landscape of multivariable functions.