Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{7 / 3}}{x^{2}+y^{2}}\)

Step-by-Step Solution

Answer

The limit does not exist because it is path-dependent.

Step 1: Analyze the Given Function

The function to analyze is \( f(x, y) = \frac{x^{7/3}}{x^2 + y^2} \). This function involves two variables, \( x \) and \( y \), and approaches the point \((0, 0)\). We need to determine whether the limit exists as \((x, y)\) approaches \((0, 0)\).

Step 2: Evaluate Along the x-axis

On the x-axis, \( y = 0 \), so the function simplifies to \( f(x, 0) = \frac{x^{7/3}}{x^2} = x^{1/3} \). As \( x \) approaches 0, \( x^{1/3} \) approaches 0. Therefore, along the x-axis, the limit of the function is 0.

Step 3: Evaluate Along the y-axis

On the y-axis, \( x = 0 \), so the function becomes \( f(0, y) = \frac{0}{y^2} = 0 \). The limit along the y-axis is also 0 as \( y \) approaches 0.

Step 4: Evaluate Along y = x

For \( y = x \), substitute to get \( f(x, x) = \frac{x^{7/3}}{x^2 + x^2} = \frac{x^{7/3}}{2x^2} = \frac{x^{-5/3}}{2} \). As \( x \) approaches 0, \( x^{-5/3} \) becomes undefined because it tends to infinity. Thus, along this path, the limit does not exist.

Step 5: Conclude Based on Path Dependency

Since the limit along different paths (like the x-axis and y = x) gave inconsistent results - one path tending to a finite limit and another tending to infinity - the limit \( \lim_{(x, y) \rightarrow(0,0)} \frac{x^{7/3}}{x^{2}+y^{2}} \) does not exist.

Key Concepts

Path Dependency

When dealing with multivariable functions, understanding the concept of path dependency is essential. In a multivariable function, a limit might differ depending on the path taken towards a specific point. This is similar to walking towards a destination using different routes and arriving at distinct locations.
To check for path dependency, consider different approaches to the point of interest. For example:

• Approaching along the x-axis (where y=0)
• Approaching along the y-axis (where x=0)
• Approaching along other curves (like y=x)

If the limits along these paths yield different results, the multivariable limit doesn't exist. The disagreement in limit values along various paths, as seen in the solution, indicates path dependency. This consistency check is crucial to ascertain whether a limit truly exists in the multivariable setting. By examining "path dependency," one can better grasp the complexity involved in deciding limits and the impact of multidirectional approaches.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of more than one variable. Think of it as stepping from a one-dimensional line into a two-dimensional plane or even beyond. When dealing with multiple variables, new challenges, such as evaluating limits in multiple directions, arise.
This branch of calculus allows us to explore varying rates of change and how multivariable functions behave in different dimensions. It creates tools for understanding phenomena across various disciplines, such as physics and engineering. Key components include partial derivatives, multiple integrals, and the study of surface and volume - all aiming to handle more complex behaviors than in single-variable situations.

• Evaluate functions with respect to more than one variable, considering each independently and jointly.
• Employ graphical, numerical, and analytical methods to gain a deeper understanding of complex relationships.
• Utilize different strategies (e.g., paths) to evaluate limits and establish continuity.

Grasping multivariable calculus enables students to analyze and interpret situations that would otherwise be overly complex in a single-variable context.

Limit Evaluation

The evaluation of limits is pivotal in both single-variable and multivariable calculus, serving as a tool for understanding behaviors near a point of interest. To perform limit evaluation in multivariable functions, it's vital to explore various paths and determine if a consistent limit value ensues. Generally, the steps for evaluating limits in multivariable functions involve:

• Identifying the function's behavior as the variables approach the targeted point.
• Trying different paths or curves leading to the point to see if they provide consistent limit results.
• Recognizing that if inconsistent results are obtained, the limit does not exist.

To illustrate: - Along the x-axis and y-axis, the given problem's limit is found to be 0. - Along the line y=x, the limit becomes undefined as it tends toward infinity. The discrepancy in results confirms that the limit of the multivariable function does not exist when approached from different paths. This methodical approach to evaluating limits is crucial in accurately describing the behavior of complex functions at specific points, ultimately providing a deeper understanding and insight into more advanced calculus topics.