Problem 10 - The Calculus with Analytic Geometry

Question

In Exercises 7 through 12, prove that for the given function \(f, \lim _{(x, y) \rightarrow(0,0)} f(x, y)\) does not exist. \(f(x, y)=\frac{x^{4}+3 x^{2} y^{2}+2 x y^{3}}{\left(x^{2}+y^{2}\right)^{2}}\)

Step-by-Step Solution

Answer

The limit does not exist because it yields different values along different paths.

Step 1: Understand the Function

The given function is \( f(x, y) = \frac{x^{4} + 3 x^{2} y^{2} + 2 x y^{3}}{(x^{2} + y^{2})^{2}} \). The goal is to prove that \( \lim _{(x, y) \rightarrow (0, 0)} f(x, y) \) does not exist.

Step 2: Analyse along the x-axis

First, substitute \( y = 0 \) and analyze the limit as \( x \rightarrow 0 \):\[ f(x, 0) = \frac{x^{4}}{x^{4}} = 1 \]This simplifies to 1.

Step 3: Analyse along the y-axis

Next, substitute \( x = 0 \) and analyze the limit as \( y \rightarrow 0 \):\[ f(0, y) = \frac{0 + 0 + 0}{y^{4}} = 0 \]This simplifies to 0.

Step 4: Analyse along the line y = x

Now, substitute \( y = x \) and analyze the limit as \( x \rightarrow 0 \):\[ f(x, x) = \frac{x^{4} + 3 x^{4} + 2 x^{4}}{(x^{2} + x^{2})^{2}} = \frac{6 x^{4}}{4 x^{4}} = \frac{3}{2} \]This simplifies to \( \frac{3}{2} \).

Step 5: Conclusion of Different Limits

Since the values of the limits along different paths (\(1\) along the x-axis, \(0\) along the y-axis, and \(\frac{3}{2}\) along y = x) are not equal, the limit \( \lim_{(x,y)\rightarrow(0,0)} f(x,y) \) does not exist.

Key Concepts

Limit Analysis

Limit analysis is a crucial concept in calculus that determines how a function approaches a certain value. When dealing with multivariable functions, analyzing limits becomes more complex. You must consider multiple pathways to the point of interest. In this exercise, we examined the limit of the function ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline ewlineewline at the point (0,0) by evaluating it along different paths: the x-axis, the y-axis, and the line y = x.ewline Consistent results along all paths indicate the limit exists. Differing results, like in our example, show the limit does not exist.ewline Important tips: For limit analysis in multivariable functions:ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline Calculate limits for different paths. Non-equal results confirm undefined limits.ewline Evaluate simpler paths first, like the axes, for easier computation.ewline Use more paths if initial assessments are inconclusive.

Path-Dependent Limits

Path-dependent limits occur when the value of a limit varies based on the chosen path toward the point. For instance, in our exercise, evaluating the function ewlineewlineewlineewline along different paths (x, y) -> (0,0) yielded different results: ewline y = 0 ewlineewline x = 0ewline y = xewline ewlineewline means exploring various trajectories to the same destination. The main takeaway is: ewlineewline Consider multiple paths for accurate multivariable limits.ewline Interpret differing results as proof of the limit's nonexistence.ewline Envision additional paths for complex functions, such as y = mx or y = x^2. Applying these techniques ensures a rigorous analysis of multivariable limits.

Multivariable Calculus

Multivariable calculus extends basic calculus to functions with multiple variables. It encompasses limits, derivatives, and integrals for functions in higher dimensions. ewline Our exercise focused on limits, a core topic in multivariable calculus. This domain heavily relies on understanding the behavior of functions when inputs approaching a certain point.ewline When working through limits in multivariable calculus:ewline Understand the function's nature and its components.ewline Evaluate simple linear paths first, then explore more complex paths.ewline Recognize that differing path results indicate an undefined limit.ewlineewlineewline Practical implications include multivariable calculus applications in physics, engineering, and economics. ewlineewlineewline By mastering multivariable limits, you'll build the foundation for advanced topics like partial differentiation and multiple integrals. Keep practicing multi-path analysis to deepen your understanding and proficiency in this essential area of calculus.