Problem 84 - Calculus Volume 3

Question

For the following exercises, evaluate the limits of the functions of three variables. $$ \lim _{(x, y, z) \rightarrow(1,2,3)} \frac{x z^{2}-y^{2} z}{x y z-1}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{3}{5}\).

Step 1: Substitute Directly

First, try substituting the values \((x, y, z) = (1, 2, 3)\) directly into the limit expression and see if it yields a determinate form. Substitute into the numerator: \(1 \cdot 3^2 - 2^2 \cdot 3 = 9 - 12 = -3\). Substitute into the denominator: \(1 \cdot 2 \cdot 3 - 1 = 6 - 1 = 5\). This results in the expression \(\frac{-3}{5}\), a valid and determinate form.

Step 2: Attempt direct substitution

Try substituting the value the variable approaches directly into the expression.

Step 3: Handle indeterminate forms

If direct substitution gives an indeterminate form, apply L'Hopital's Rule, algebraic manipulation, or other techniques.

Step 4: Evaluate the limit

Compute the final value of the limit.

Step 5: State the conclusion

Express the final answer.

Key Concepts

Multivariable Calculus

Multivariable Calculus is an extension of single-variable calculus to functions of several variables. While single-variable calculus deals with functions of one variable, multivariable calculus focuses on more complex functions involving multiple variables such as 2D or 3D coordinates. In this context, it enables us to analyze and understand phenomena in higher dimensions, which is essential in fields like physics and engineering.
When analyzing functions of several variables, we often have to evaluate limits and derivatives that depend on more than one variable. This can involve either partial derivatives or optimization techniques, among others. These analyses allow us to understand how changes in one variable affect others, and how this interconnection impacts the function as a whole.

Limits in Three Variables

Limits in three variables involve approaching a particular point in a three-dimensional space, generally represented as \(x, y, z\). The goal is to find out what happens to the function's output as the variables approach these conditions simultaneously. This is similar to two-variable limits but exists in a higher-dimensional space.
Finding such limits requires understanding paths of approach—paths like straight lines, curves, or surfaces—that lead to the point of interest. The process can be simple or require more advanced techniques if direct substitution does not work, often seen as a crucial step in verifying the consistency of the limit across all approaches.

Evaluating Limits

Evaluating limits of functions with multiple variables is much like evaluating one-variable limits, but in a multidimensional space with additional complexities. The first step typically involves direct substitution of the point to see if the limit exists, as was done in the original solution.

• If direct substitution gives a determinate form, the limit is immediate.
• If the form is indeterminate, alternative methods, such as rearranging the expression, might be needed.
• In some cases, employing paths or using analytic approaches can offer insight.

The key is to explore whether the limit remains consistent across all potential paths of approach to the point in question.

Mathematical Substitution

Mathematical substitution is a fundamental technique used to simplify complex expressions by directly replacing variables with specific values. This is often the initial approach in limit problems to ascertain whether further evaluation is necessary.
In the given exercise, substitution was used effectively by placing \(x = 1\), \(y = 2\), and \(z = 3\) directly into both the numerator and the denominator. The substitution simplified the function to a fraction \(\frac{-3}{5}\), providing a straightforward and determinate result without requiring more complex methods.
This method is both efficient and straightforward, allowing a quick check for limits' existence.