Problem 40 - Calculus with Concepts in Calculus

Question

The function \(f\) defined by $$ f(x, y)=\frac{100 a x}{a x+b y} \quad \text { for } x>0 \text { and } y>0 $$ where \(a\) and \(b\) are positive constants, appears in the study of the relationship of blood flow through the right lung to the total blood flow in the system. Determine whether \(f\) has a limit at \((0,0)\).

Step-by-Step Solution

Answer

The function \(f(x, y)\) does not have a limit at \((0,0)\) because it yields different limits along different paths.

Step 1: Understand the Problem

We are given a function \( f(x, y) = \frac{100ax}{ax + by} \) and asked to determine if it has a limit as \((x, y)\) approaches \((0, 0)\). The problem is evaluating a multivariable limit.

Step 2: Analyze the Behavior on the x-axis

To analyze the behavior along the x-axis, consider the line \(y=0\). Substitute \( y = 0 \) into the function: \[f(x, 0) = \frac{100ax}{ax} = 100 \]Thus, along the x-axis, the function approaches 100 as \( x \rightarrow 0 \).

Step 3: Analyze the Behavior on the y-axis

Now, consider the line \(x=0\). Substitute \( x = 0 \) into the function:\[f(0, y) = \frac{100 \cdot 0}{0 + by} = 0 \]Thus, along the y-axis, the function approaches 0 as \( y \rightarrow 0 \).

Step 4: Test Along Another Path

To further check the limit doesn't exist, consider the path \(y = x\). Substitute \( y = x \) in the function:\[f(x, x) = \frac{100ax}{ax + bx} = \frac{100ax}{x(a + b)} = \frac{100a}{a + b} \]Along this line, the function approaches \(\frac{100a}{a+b}\) as \(x \rightarrow 0\).

Step 5: Conclusion on the Limit

Since the limits along different paths toward \((0,0)\) yield different results — 100 along the x-axis, 0 along the y-axis, and \(\frac{100a}{a+b}\) along \(y = x\) — the function \( f \) does not have a unique limit at \((0,0)\).

Key Concepts

Functions of Two Variables

A function of two variables, often denoted as \( f(x, y) \), is a rule that assigns a single numerical output for every pair of inputs \( x \) and \( y \). In this context, we're looking at how the function behaves as we change these inputs. This is akin to investigating how a surface changes not just in one direction, but across a plane comprising both directions.

• Such functions are crucial in multivariable calculus, a branch of calculus that involves two or more variables simultaneously.
• They are often used to model real-world phenomena, for example, how pressure or temperature might change across a surface.
• In our exercise, the function \( f(x, y) = \frac{100ax}{ax + by} \) models blood flow, depending on variables \( x \) and \( y \).

Understanding how this two-variable function behaves as the inputs change allows us to make predictions or analyze behaviors in physical systems. It's like observing how changing two knobs can alter the position of a point on a plane.

Paths of Approach

When evaluating limits in multiple variables, the path of approach is a central idea. This means examining the function as it approaches a particular point along different paths. Since we have more than one variable, there are multiple directions or paths through which we can approach a point, such as \( (0, 0) \) in our exercise.

• These paths can include the x-axis, y-axis, or any line like \( y = x \), as tested in our solution.
• Examining different paths helps to determine if a unique limit exists at a point.
• If different paths give different limit values, then the limit at that point does not exist.

In our exercise, we observed different limits along the x-axis, y-axis, and another path \( (y = x) \). This variety in limits highlights why evaluating the path of approach is crucial when deciding if a multiple-variable limit exists.

Limit Evaluation

The process of limit evaluation in multivariable calculus involves determining if a function approaches a particular value as the inputs approach a specific point. Unlike single-variable calculus, we must consider multiple paths of approach, as there are infinitely many directions to a point in a plane.

• The solution first evaluated the limit along the x-axis and y-axis, each giving different results.
• We then checked the path \( y = x \) which produced yet another limiting value.
• When multiple paths yield different limits, the limit does not exist.

For our given function \( f(x, y) \), the limits vary based on the path taken. This lack of uniformity across paths signals that there is no well-defined limit at the point \( (0, 0) \). This step is critical because, in multivariable contexts, confirming a non-existent limit can indicate discontinuities or anomalies in real-world applications.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with more than one variable, like \( f(x, y) \). It allows mathematicians and scientists to explore how functions behave in higher dimensions. The study involves derivatives, integrals, and limits of such functions.

• Multivariable calculus is important in fields like physics, engineering, and economics, where variables are interdependent.
• It requires understanding how multiple variables influence the output and how the function behaves across a multidimensional surface.
• Concepts like the gradient, divergence, and curl build upon these ideas to describe phenomena in fields and fluids.

In our problem, multivariable calculus helped evaluate how the function \( f(x, y) \) behaves as both \( x \) and \( y \) approach zero. Not only does this investigation clarify mathematical properties, but it also has practical applications, such as understanding the dynamics of blood flow.