Problem 39 - Calculus with Concepts in Calculus

Question

When \(x\) moles of sulfuric acid are mixed with \(y\) moles of water, the heat \(Q(x, y)\) produced is given by $$ Q(x, y)=\frac{17,860 x y}{(1.798) x+y} \quad \text { for } x>0 \text { and } y>0 $$ Determine whether \(Q\) has a limit at \((0,0)\), and if so, compute its value.

Step-by-Step Solution

Answer

The limit of \(Q(x, y)\) at \((0,0)\) is zero.

Step 1: Analyze the function behavior near (0,0)

The function \(Q(x, y) = \frac{17860xy}{1.798x + y}\) expresses the heat produced. To find the limit at point \((0,0)\), we need to analyze what happens when both \(x\) and \(y\) approach zero at the same time.

Step 2: Direct substitution check

Attempt to substitute \(x = 0\) and \(y = 0\) directly into the function. Doing so gives \(Q(0, 0) = \frac{17860(0)(0)}{1.798(0) + 0} = \frac{0}{0}\), which is undefined. This requires further analysis.

Step 3: Approach along specific paths

Let's try finding the limit of \(Q(x, y)\) as \((x, y)\) approaches \((0,0)\) along specific paths. First, substitute \(y = kx\), where \(k\) is a constant, giving \[Q(x, kx) = \frac{17860x(kx)}{1.798x + kx} = \frac{17860kx^2}{x(1.798 + k)} = \frac{17860kx}{1.798 + k}\].

Step 4: Compute the limit along each path

Calculate \(\lim_{x \to 0} Q(x, kx) = \lim_{x \to 0} \frac{17860kx}{1.798 + k} = 0\) since \(x\) is in the numerator and goes to zero.

Step 5: Further examination of path independence

To ensure the limit is path-independent, try another path, say \(x = ky\). Substitute this into the function: \[Q(ky, y) = \frac{17860(ky)y}{1.798(ky) + y} = \frac{17860ky^2}{y(1.798k + 1)} = \frac{17860ky}{1.798k + 1}\]. Taking the limit as \(y \to 0\), we also get zero: \(\lim_{y \to 0} Q(ky, y) = 0\).

Step 6: Conclude the limit

Both paths resulted in the limit being zero and show the function behaves similarly regardless of the path chosen as \(x\) and \(y\) approach zero. This indicates \(\lim_{(x, y) \to (0,0)} Q(x, y) = 0\), and therefore the limit exists and is zero.

Key Concepts

Understanding Multivariable Functions

Multivariable functions, like the one in this problem, depend on more than one variable. In our example, the function \( Q(x, y) \) represents the heat produced when \( x \) moles of sulfuric acid are mixed with \( y \) moles of water.

The way it works is simple: both \( x \) and \( y \) influence the outcome of \( Q(x, y) \), meaning that the function takes both of these inputs and provides a single output — the amount of heat produced. Such functions help to study the relationships between different quantities.

Understanding these functions involves examining how changes in the variables affect the output, which is crucial for grasping how to work with limits in multivariable calculus. This builds up a foundational tool for analyzing changes and behaviors in different fields of study.

Exploring Path Independence

Path independence is an essential concept to understand when dealing with multivariable limits. It means that when you're finding the limit of a function like \( Q(x, y) \) at a particular point, say \((0,0)\), the final value you arrive at should not depend on the path taken by \( (x, y) \) as it approaches \((0,0)\).

In planar cases, this means trying different paths, such as lines or curves, and checking if the limit of the function along each path leads to the same result. If they do, as in our example with \( Q(x, y) \), the limit is said to be path-independent.

• For instance, in our solution, substituting \( y = kx \) and calculating the limit showed that it resolved to 0.
• Similarly, substituting \( x = ky \) also led to the limit being 0.

This consistent result across different paths affirms that \( Q(x, y) \) indeed approaches 0 as \( (x, y) \) draws closer to \((0,0)\), validating our solution.

Understanding Limit Calculations in Multivariable Calculus

Limit calculations in multivariable calculus can be more intricate than those in single-variable calculus because they involve multiple paths of approach.

In the exercise, the function \( Q(x, y) \) needed to be evaluated as \((x,y)\) approached \((0,0)\). Initially, direct substitution led to an indeterminate form \( \frac{0}{0} \). That signals us to explore further rather than stopping.

By calculating the limit along different paths, such as \( y = kx \) or \( x = ky \), we obtain results that help verify the limit value. We followed these steps and found that along each path, the limit still equated to 0, indicating that \( Q(x, y) \) safely approaches 0.

• This means the limit indeed exists, and it is path-independent.
• Thus, the calculated limit of \( Q(x, y) \) at \((0,0)\) is zero.

These insights showcase the beauty and challenge of limits in multivariable calculus, emphasizing thorough evaluations along multiple paths.